* 


A  PHILOSOPHICAL  ESSAY 

ON 

PROBABILITIES. 

BY 

PIERRE  SIMON,   MARQUIS  DE  LAPLACE. 

TRANSLATED  FROM  THE  SIXTH  FRENCH  EDITION 


FREDERICK  WILSON  TRUSCOTT,  PH.D.  (HARV.), 

Professor  of  Germanic  Languages  in  the  U'est  Virginia.  University, 


FREDERICK  LINCOLN  EMORY,  M.E.  (WoR.  POLY.  INST.), 

Professor  of  Mechanics  and  Applied  Mathematics  in  the    West  Virginia 
University  ;  Mem.  Amer.  Soc.  Mtch.  Eng. 


FIRST  EDITION. 
FIRST    THOUSAND. 


NEW  YORK: 

JOHN  WILEY  &  SONS. 

LONDON  :  CHAPMAN  &  HALL,  LIMITED. 

1902. 


Copyright,  1902, 

BY 

F.  W.  TRUSCOTT 
F.  L.  EMORY. 


ROBERT    DRUMMOND      PRINTER,    NEW   YORK 


Stack 
Annex 


TABLE  OF  CONTENTS. 


PART  I. 

A   PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 
CHAPTER  I. 

PAGE 

Introduction i 

CHAPTER  II. 
Concerning  Probability 3 

CHAPTER  III. 
General  Principles  of  the  Calculus  of  Probabilities 1 1 

CHAPTER  IV. 
Concerning  Hope 20 

CHAPTER  V. 
Analytical  Methods  of  the  Calculus  of  Probabilities 26 


PART  II. 

APPLICATION  OF   THE  CALCULUS  OF 
PROBABILITIES. 

CHAPTER  VI. 
Games  of  Chance 53 

CHAPTER  VII. 

Concerning    the  Unknown    Inequalities  which   may    Exist  among 

Chances  Supposed  to  be  Equal 56 

iii 


iv  TABLE  OF  CONTENTS. 

CHAPTER  VIII. 

PACK 

Concerning  the  Laws  of  Probability  which  result  from  the  Indefinite 

Multiplication  of  Events 6° 

CHAPTER  IX. 
Application  of  the  Calculus  of  Probabilities  to  Natural  Philosophy. .     73 

CHAPTER  X. 
Application  of  the  Calculus  of  Probabilities  to  the  Moral  Sciences. .    107 

CHAPTER  XL 
Concerning  the  Probability  of  Testimonies 109 

CHAPTER  XII. 
Concerning  the  Selections  and  Decisions  of  Assemblies 126 

CHAPTER  XIII. 
Concerning  the  Probability  of  the  Judgments  of  Tribunals 132 

CHAPTER  XIV. 
Concerning  Tables  of  Mortality,  and  the  Mean  Durations  of  Life, 

Marriage,  and  Some  Associations 140 

CHAPTER  XV. 
Concerning  the  Benefits  of  Institutions  which  Depend  upon  the 

Probability  of  Events 149 

CHAPTER  XVI. 
Concerning  Illusions  in  the  Estimation  of  Probabilities 160 

CHAPTER  XVII. 
Concerning  the  Various  Means  of  Approaching  Certainty 176 

CHAPTER  XVIII. 
Historical  Notice  of  the  Calculus  of  Probabilities  to  1816 185 


ERRATA. 

Page  89,  line    22,          for         Pline          read          Pliny 

"    102,  lines  14,  16,      "  minutes         "  days 

"    143,  line    25,  "          sun  soil 

"    177,  lines  15,  17,  18,  21,  22,  24,  for  primary  read  prime 
"    182,  line      5,          for     conjunctions     read    being  binary 


A  PHILOSOPHICAL   ESSAY   ON 
PROBABILITIES. 


CHAPTER   I. 
INTRODUCTION. 

THIS  philosophical  essay  is  the  development  of  a 
lecture  on  probabilities  which  I  delivered  in  1795  to 
the  normal  schools  whither  I  had  been  called,  by  a 
decree  of  the  national  convention,  as  professor  of 
mathematics  with  Lagrange.  I  have  recently  published 
upon  the  same  subject  a  work  entitled  The  Analytical 
Theory  of  Probabilities.  I  present  here  without  the 
aid  of  analysis  the  principles  and  general  results  of  this 
theory,  applying  them  to  the  most  important  questions 
of  life,  which  are  indeed  for  the  most  part  only  problems 
of  probability.  Strictly  speaking  it  may  even  be  said 
that  nearly  all  our  knowledge  is  problematical ;  and  in 
the  small  number  of  things  which  we  are  able  to  know 
with  certainty,  even  in  the  mathematical  sciences 
themselves,  the  principal  means  for  ascertaining  truth 
— induction  and  analogy — are  based  on  probabilities; 


2          A  PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 

so  that  the  entire  system  of  human  knowledge  is  con- 
nected with  the  theory  set  forth  in  this  essay.  Doubt- 
less it  will  be  seen  here  with  interest  that  in  considering, 
even  in  the  eternal  principles  of  reason,  justice,  and 
humanity,  only  the  favorable  chances  which  are  con- 
stantly attached  to  them,  there  is  a  great  advantage  in 
following  these  principles  and  serious  inconvenience  in 
departing  from  them:  their  chances,  like  those  favor- 
able to  lotteries,  always  end  by  prevailing  in  the  midst 
of  the  vacillations  of  hazard.  I  hope  that  the  reflec- 
tions given  in  this  essay  may  merit  the  attention  of 
philosophers  and  direct  it  to  a  subject  so  worthy  of 
engaging  their  minds. 


CHAPTER    II. 
CONCERNING  PROBABILITY. 

ALL  events,  even  those  which  on  account  of  their 
insignificance  do  not  seem  to  follow  the  great  laws  of 
nature,  are  a  result  of  it  just  as  necessarily  as  the  revolu- 
tions of  the  sun.  In  ignorance  of  the  ties  which  unite 
such  events  to  the  entire  system  of  the  universe,  they 
have  been  made  to  depend  upon  final  causes  or  upon 
hazard,  according  as  they  occur  and  are  repeated  with 
regularity,  or  appear  without  regard  to  order ;  but  these 
imaginary  causes  have  gradually  receded  with  the 
widening  bounds  of  knowledge  and  disappear  entirely 
before  sound  philosophy,  which  sees  in  them  only  the 
expression  of  our  ignorance  of  the  true  causes. 

Present  events  are  connected  with  preceding  ones 
by  a  tie  based  upon  the  evident  principle  that  a  thing 
cannot  occur  without  a  cause  which  produces  it.  This 
axiom,  known  by  the  name  of  the  principle  of  sufficient 
reason,  extends  even  to  actions  which  are  considered 
indifferent ;  the  freest  will  is  unable  without  a  determi- 
native motive  to  give  them  birth ;  if  we  assume  two 
positions  with  exactly  similar  circumstances  and  find 
that  the  will  is  active  in  the  one  and  inactive  in  the 

3 


4          A  PHILOSOPHICAL  BBS  AY  ON  PROBABILITIES. 

other,  we  say  that  its  choice  is  an  effect  without  a  cause. 
It  is  then,  says  Leibnitz,  the  blind  chance  of  the 
Epicureans.  The  contrary  opinion  is  an  illusion  of  the 
mind,  which,  losing  sight  of  the  evasive  reasons  of  the 
choice  of  the  will  in  indifferent  things,  believes  that 
choice  is  determined  of  itself  and  without  motives. 

We  ought  then  to  regard  the  present  state  of  the 
universe  as  the  effect  of  its  anterior  state  and  as  the 
cause  of  the  one  which  is  to  follow.  Given  for  one 
instant  an  intelligence  which  could  comprehend  all  the 
forces  by  which  nature  is  animated  and  the  respective 
situation  of  the  beings  who  compose  it — an  intelligence 
sufficiently  vast  to  submit  these  data  to  analysis — it 
would  embrace  in  the  same  formula  the  movements  of 
the  greatest  bodies  of  the  universe  and  those  of  the 
lightest  atom ;  for  it,  nothing  would  be  uncertain  and 
the  future,  as  the  past,  would  be  present  to  its  eyes. 
The  human  mind  offers,  in  the  perfection  which  it  has 
been  able  to  give  to  astronomy,  a  feeble  idea  of  this  in- 
telligence. Its  discoveries  in  mechanics  and  geometry, 
added  to  that  of  universal  gravity,  have  enabled  it  to 
comprehend  in  the  same  analytical  expressions  the 
past  and  future  states  of  the  system  of  the  world. 
Applying  the  same  method  to  some  other  objects  of  its 
knowledge,  it  has  succeeded  in  referring  to  general  laws 
observed  phenomena  and  in  foreseeing  those  which 
given  circumstances  ought  to  produce.  All  these  efforts 
in  the  search  for  truth  tend  to  lead  it  back  continually 
to  the  vast  intelligence  which  we  have  just  mentioned, 
but  from  which  it  will  always  remain  infinitely  removed. 
This  tendency,  peculiar  to  the  human  race,  is  that 
which  renders  it  superior  to  animals ;  and  their  progress 


CONCERNING   PROBABILITY.  5 

in  this  respect  distinguishes  nations  and  ages  and  con- 
stitutes their  true  glory. 

Let  us  recall  that  formerly,  and  at  no  remote  epoch, 
an  unusual  rain  or  an  extreme  drought,  a  comet  having 
in  train  a  very  long  tail,  the  eclipses,  the  aurora 
borealis,  and  in  general  all  the  unusual  phenomena 
were  regarded  as  so  many  signs  of  celestial  wrath. 
Heaven  was  invoked  in  order  to  avert  their  baneful 
influence.  No  one  prayed  to  have  the  planets  and  the 
sun  arrested  in  their  courses:  observation  had  soon 
made  apparent  the  futility  of  such  prayers.  But  as 
these  phenomena,  occurring  and  disappearing  at  long 
intervals,  seemed  to  oppose  the  order  of  nature,  it  was 
supposed  that  Heaven,  irritated  by  the  crimes  of  the 
earth,  had  created  them  "to  announce  its  vengeance. 
Thus  the  long  tail  of  the  comet  of  1456  spread  terror 
through  Europe,  already  thrown  into  consternation  by 
the  rapid  successes  of  the  Turks,  who  had  just  over- 
thrown the  Lower  Empire.  This  star  after  four  revolu- 
tions has  excited  among  us  a  very  different  interest. 
The  knowledge  of  the  laws  of  the  system  of  the  world 
acquired  in  the  interval  had  dissipated  the  fears 
begotten  by  the  ignorance  of  the  true  relationship  of 
man  to  the  universe;  and  Halley,  having  recognized 
the  identity  of  this  comet  with  those  of  the  years  1531, 
1607,  and  1682,  announced  its  next  return  for  the  end 
of  the  year  1758  or  the  beginning  of  the  year  1759. 
The  learned  world  awaited  with  impatience  this  return 
which  was  to  confirm  one  of  the  greatest  discoveries 
that  have  been  made  in  the  sciences,  and  fulfil  the 
prediction  of  Seneca  when  he  said,  in  speaking  of  the 
revolutions  of  those  stars  which  fall  from  an  enormous 


6          A  PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 

height:  "The  day  will  come  when,  by  study  pursued 
through  several  ages,  the  things  now  concealed  will 
appear  with  evidence;  and  posterity  will  be  astonished 
that  truths  so  clear  had  escaped  us. ' '  Clairaut  then 
undertook  to  submit  to  analysis  the  perturbations  which 
the  comet  had  experienced  by  the  action  of  the  two 
great  planets,  Jupiter  and  Saturn;  after  immense  cal- 
culations he  fixed  its  next  passage  at  the  perihelion 
toward  the  beginning  of  April,  1759,  which  was  actually 
verified  by  observation.  The  regularity  which  astronomy 
shows  us  in  the  movements  of  the  comets  doubtless 
exists  also  in  all  phenomena.  - 

The  curve  described  by  a  simple  molecule  of  air  or 
vapor  is  regulated  in  a  manner  just  as  certain  as  the 
planetary  orbits ;  the  only  difference  between  them  is 
that  which  comes  from  our  ignorance. 

Probability  is  relative,  in  part  to  this  ignorance,  in 
part  to  our  knowledge.  We  know  that  of  three  or  a 
greater  number  of  events  a  single  one  ought  to  occur ; 
but  nothing  induces  us  to  believe  that  one  of  them  will 
occur  rather  than  the  others.  In  this  state  of  indecision 
it  is  impossible  for  us  to  announce  their  occurrence  with 
certainty.  It  is,  however,  probable  that  one  of  these 
events,  chosen  at  will,  will  not  occur  because  we  see 
several  cases  equally  possible  which  exclude  its  occur- 
rence, while  only  a  single  one  favors  it. 

The  theory  of  chance  consists  in  reducing  all  the 
events  of  the  same  kind  to  a  certain  number  of  cases 
equally  possible,  that  is  to  say,  to  such  as  we  may  be 
equally  undecided  about  in  regard  to  their  existence, 
and  in  determining  the  number  of  cases  favorable  to 
the  event  whose  probability  is  sought.  The  ratio  of 


CONCERNING  PROBABILITY.  7 

this  number  to  that  of  all  the  cases  possible  is  the 
measure  of  this  probability,  which  is  thus  simply  a 
fraction  whose  numerator  is  the  number  of  favorable 
cases  and  whose  denominator  is  the  number  of  all  the 
cases  possible. 

The  preceding  notion  of  probability  supposes  that, 
in  increasing  in  the  same  ratio  the  number  of  favorable 
cases  and  that  of  all  the  cases  possible,  the  probability 
remains  the  same.  In  order  to  convince  ourselves  let 
us  take  two  urns,  A  and  B,  the  first  containing  four 
white  and  two  black  balls,  and  the  second  containing 
only  two  white  balls  and  one  black  one.  We  may 
imagine  the  two  black  balls  of  the  first  urn  attached  by 
a  thread  which  breaks  at  the  moment  when  one  of 
them  is  seized  in  order  to  be  drawn  out,  and  the  four 
white  balls  thus  forming  two  similar  systems.  All  the 
chances  which  will  favor  the  seizure  of  one  of  the  balls 
of  the  black  system  will  lead  to  a  black  ball.  If  we 
conceive  now  that  the  threads  which  unite  the  balls  do 
not  break  at  all,  it  is  clear  that  the  number  of  possible 
chances  will  not  change  any  more  than  that  of  the 
chances  favorable  to  the  extraction  of  the  black  balls; 
but  two  balls  will  be  drawn  from  the  urn  at  the  same 
time ;  the  probability  of  drawing  a  black  ball  from  the 
urn  A  will  then  be  the  same  as  at  first.  But  then  we 
have  obviously  the  case  of  urn  B  with  the  single  differ- 
ence that  the  three  balls  of  this  last  urn  would  be 
replaced  by  three  systems  of  two  balls  invariably  con- 
nected. 

When  all  the  cases  are  favorable  to  an  event  the 
probability  changes  to  certainty  and  its  expression 
becomes  equal  to  unity.  Upon  this  condition,  certainty 


8          A  PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 

and  probability  are  comparable,  although  there  may  be 
an  essential  difference  between  the  two  states  of  the 
mind  when  a  truth  is  rigorously  demonstrated  to  it,  cr 
when  it  still  perceives  a  small  source  of  error. 

In  things  which  are  only  probable  the  difference  of 
the  data,  which  each  man  has  in  regard  to  them,  is  one 
of  the  principal  causes  of  the  diversity  of  opinions  which 
prevail  in  regard  to  the  same  objects.  Let  us  suppose, 
for  example,  that  we  have  three  urns,  A,  B,  C,  one  of 
which  contains  only  black  balls  while  the  two  others 
contain  only  white  balls ;  a  ball  is  to  be  drawn  from 
the  urn  C  and  the  probability  is  demanded  that  this 
ball  will  be  black.  If  we  do  not  know  which  of  the 
three  urns  contains  black  balls  only,  so  that  there  is  no 
reason  to  believe  that  it  is  C  rather  than  B  or  A,  these 
three  hypotheses  will  appear  equally  possible,  and  since 
a  black  ball  can  be  drawn  only  in  the  first  hypothesis, 
the  probability  of  drawing  it  is  equal  to  one  third.  If 
it  is  known  that  the  urn  A  contains  white  balls  only, 
the  indecision  then  extends  only  to  the  urns  B  and  C, 
and  the  probability  that  the  ball  drawn  from  the  urn  C 
will  be  black  is  one  half.  Finally  this  probability 
changes  to  certainty  if  we  are  assured  that  the  urns  A 
and  B  contain  white  balls  only. 

It  is  thus  that  an  incident  related  to  a  numerous 
assembly  finds  various  degrees  of  credence,  according 
to  the  extent  of  knowledge  of  the  auditors.  If  the 
man  who  reports  it  is  fully  convinced  of  it  and  if,  by 
his  position  and  character,  he  inspires  great  confidence, 
his  statement,  however  extraordinary  it  may  be,  will 
have  for  the  auditors  who  lack  information  the  same 
degree  of  probability  as  an  ordinary  statement  made 


CONCERNING   PROBABILITY.  9 

by  the  same  man,  and  they  will  have  entire  faith  in  it. 
But  if  some  one  of  them  knows  that  the  same  incident 
is  rejected  by  other  equally  trustworthy  men,  he  will 
be  in  doubt  and  the  incident  will  be  discredited  by  the 
enlightened  auditors,  who  will  reject  it  whether  it  be 
in  regard  to  facts  well  averred  or  the  immutable  laws 
of  nature. 

It  is  to  the  influence  of  the  opinion  of  those  whom 
the  multitude  judges  best  informed  and  to  whom  it  has 
been  accustomed  to  give  its  confidence  in  regard  to 
the  most  important  matters  of  life  that  the  propagation 
of  those  errors  is  due  which  in  times  of  ignorance  have 
covered  the  face  of  the  earth.  Magic  and  astrology 
offer  us  two  great  examples.  These  errors  inculcated 
in  infancy,  adopted  without  examination,  and  having 
for  a  basis  only  universal  credence,  have  maintained 
themselves  during  a  very  long  time ;  but  at  last  the 
progress  of  science  has  destroyed  them  in  the  minds  of 
enlightened  men,  whose  opinion  consequently  has 
caused  them  to  disappear  even  among  the  common 
people,  through  the  power  of  imitation  and  habit  wrhich 
had  so  generally  spread  them  abroad.  This  power, 
the  richest  resource  of  the  moral  world,  establishes  and 
conserves  in  a  whole  nation  ideas  entirely  contrary  to 
those  which  it  upholds  elsewhere  with  the  same 
authority.  What  indulgence  ought  we  not  then  to 
have  for  opinions  different  from  ours,  when  this  differ- 
ence often  depends  only  upon  the  various  points  of  view 
where  circumstances  have  placed  us!  Let  us  enlighten 
those  whom  we  judge  insufficiently  instructed ;  but  first 
let  us  examine  critically  our  own  opinions  and  weigh 
with  impartiality  their  respective  probabilities. 


TO        A  PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 

The  difference  of  opinions  depends,  however,  upon 
the  manner  in  which  the  influence  of  known  data  is 
determined.  The  theory  of  probabilities  holds  to  con- 
siderations so  delicate  that  it  is  not  surprising  that  with 
the  same  data  two  persons  arrive  at  different  results, 
especially  in  very  complicated  questions.  Let  us 
examine  now  the  general  principles  of  this  theory. 


CHAPTER    III. 

THE  GENERAL   PRINCIPLES  OF   THE    CALCULUS 
OF  PROBABILITIES. 

First  Principle. — The  first  of  these  principles  is  the 
definition  itself  of  probability,  which,  as  has  been  seen, 
is  the  ratio  of  the  number  of  favorable  cases  to  that  of 
all  the  cases  possible. 

Second  Principle. — But  that  supposes  the  various 
cases  equally  possible.  If  they  are  not  so,  we  will 
determine  first  their  respective  possibilities,  whose 
exact  appreciation  is  one  of  the  most  delicate  points  of 
the  theory  of  chance.  Then  the  probability  will  be 
the  sum  of  the  possibilities  of  each  favorable  case. 
Let  us  illustrate  this  principle  by  an  example. 

Let  us  suppose  that  we  throw  into  the  air  a  large 
and  very  thin  coin  whose  two  large  opposite  faces, 
which  we  will  call  heads  and  tails,  are  perfectly  similar. 
Let  us  find  the  probability  of  throwing  heads  at  least 
one  time  in  two  throws.  It  is  clear  that  four  equally 
possible  cases  may  arise,  namely,  heads  at  the  first 
and  at  the  second  throw ;  heads  at  the  first  throw  and 
tails  at  the  second;  tails  at  the  first  throw  and  heads 
at  the  second;  finally,  tails  at  both  throws.  The  first 


12        A  PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

three  cases  are  favorable  to  the  event  whose  probability 
is  sought;  consequently  this  probability  is  equal  to  |; 
so  that  it  is  a  bet  of  three  to  one  that  heads  will  be 
thrown  at  least  once  in  two  throws. 

We  can  count  at  this  game  only  three  different  cases, 
namely,  heads  at  the  first  throw,  which  dispenses  with 
throwing  a  second  time;  tails  at  the  first  throw  and 
heads  at  the  second ;  finally,  tails  at  the  first  and  at  the 
second  throw.  This  would  reduce  the  probability  to 
|  if  we  should  consider  with  d'Alembert  these  three 
cases  as  equally  possible.  But  it  is  apparent  that  the 
probability  of  throwing  heads  at  the  first  throw  is  f , 
while  that  of  the  two  other  cases  is  J,  the  first  case 
being  a  simple  event  which  corresponds  to  two  events 
combined :  heads  at  the  first  and  at  the  second  throw, 
and  heads  at  the  first  throw,  tails  at  the  second.  If 
we  then,  conforming  to  the  second  principle,  add  the 
possibility  f  of  heads  at  the  first  throw  to  the  possi- 
bility J  of  tails  at  the  first  throw  and  heads  at  the 
second,  we  shall  have  f  for  the  probability  sought, 
which  agrees  with  what  is  found  in  the  supposition 
when  we  play  the  two  throws.  This  'supposition  does 
not  change  at  all  the  chance  of  that  one  who  bets  on 
this  event;  it  simply  serves  to  reduce  the  various  cases 
to  the  cases  equally  possible. 

Third  Principle. — One  of  the  most  important  points 
of  the  theory  of  probabilities  and  that  which  lends  the 
most  to  illusions  is  the  manner  in  which  these  prob- 
abilities increase  or  diminish  by  their  mutual  combina- 
tion. If  the  events  are  independent  of  one  another,  the 
probability  of  their  combined  existence  is  the  product 
of  their  respective  probabilities.  Thus  the  probability 


CALCULUS   OF  PROBABILITIES.  13 

of  throwing  one  ace  with  a  single  die  is  ^;  that  of 
throwing  two  aces  in  throwing  two  dice  at  the  same 
time  is  -£-$.  Each  face  of  the  one  being  able  to  com- 
bine with  the  six  faces  of  the  other,  there  are  in  fact 
thirty-six  equally  possible  cases,  among  which  one 
single  case  gives  two  aces.  Generally  the  probability 
that  a  simple  event  in  the  same  circumstances  will 
occur  consecutively  a  given  number  of  times  is  equal  to 
the  probability  of  this  simple  event  raised  to  the  power 
indicated  by  this  number.  Having  thus  the  successive 
powers  of  a  fraction  less  than  unity  diminishing  without 
ceasing,  an  event  which  depends  upon  a  series  of  very 
great  probabilities  may  become  extremely  improbable. 
Suppose  then  an  incident  be  transmitted  to  us  by 
twenty  witnesses  in  such  manner  that  the  first  has 
transmitted  it  to  the  second,  the  second  to  the  third, 
and  so  on.  Suppose  again  that  the  probability  of  each 
testimony  be  equal  to  the  fraction  T9¥;  that  of  the 
incident  resulting  from  the  testimonies  will  be  less 
than  £.  We  cannot  better  compare  this  diminution  of 
the  probability  than  with  the  extinction  of  the  light  of 
objects  by  the  interposition  of  several  pieces  of  glass. 
A  relatively  small  number  of  pieces  suffices  to  take 
away  the  view  of  an  object  that  a  single  piece  allows 
us  to  perceive  in  a  distinct  manner.  The  historians  do 
not  appear  to  have  paid  sufficient  attention  to  this 
degradation  of  the  probability  of  events  when  seen 
across  a  great  number  of  successive  generations;  many 
historical  events  reputed  as  certain  would  be  at  least 
doubtful  if  they  were  submitted  to  this  test. 

In  the  purely  mathematical  sciences  the  most  distant 
consequences  participate  in  the  certainty  of  the  princi- 


M        A  PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 

pie  from  which  they  are  derived.  In  the  applications 
of  analysis  to  physics  the  results  have  all  the  certainty 
of  facts  or  experiences.  But  in  the  moral  sciences, 
where  each  inference  is  deduced  from  that  which  pre- 
cedes it  only  in  a  probable  manner,  however  probable 
these  deductions  may  be,  the  chance  of  error  increases 
with  their  number  and  ultimately  surpasses  the  chance 
of  truth  in  the  consequences  very  remote  from  the 
principle. 

Fourth  Principle. — When  two  events  depend  upon 
each  other,  the  probability  of  the  compound  event  is 
the  product  of  the  probability  of  the  first  event  and  the 
probability  that,  this  event  having  occurred,  the  second 
will  occur.  Thus  in  the  preceding  case  of  the  three 
urns  A,  B,  C,  of  which  two  contain  only  white  balls 
and  one  contains  only  black  balls,  the  probability  of 
drawing  a  white  ball  from  the  urn  C  is  f ,  since  of  the 
three  urns  only  two  contain  balls  of  that  color.  But 
when  a  white  ball  has  been  drawn  from  the  urn  C,  the 
indecision  relative  to  that  one  of  the  urns  which  contain 
only  black  balls  extends  only  to  the  urns  A  and  B; 
the  probability  of  drawing  a  white  ball  from  the  urn  B 
is  £ ;  the  product  of  \  by  £,  or  £,  is  then  the  probability 
of  drawing  two  white  balls  at  one  time  from  the  urns 
B  and  C. 

We  see  by  this  example  the  influence  of  past  events 
upon  the  probability  of  future  events.  For  the  prob- 
ability of  drawing  a  white  ball  from  the  urn  B,  which 
primarily  is  f,  becomes  \  when  a  white  ball  has  been 
drawn  from  the  urn  C ;  it  would  change  to  certainty  if 
a  black  ball  had  been  drawn  from  the  same  urn.  We 
will  determine  this  influence  by  means  of  the  follow- 


CALCULUS   OF  PROBABILITIES.  15 

ing  principle,  which  is  a  corollary  of  the  preceding 
one. 

Fifth  Principle. — If  we  calculate  a  priori  the  prob- 
ability of  the  occurred  event  and  the  probability  of  an 
event  composed  of  that  one  and  a  second  one  which  is 
expected,  the  second  probability  divided  by  the  first 
will  be  the  probability  of  the  event  expected,  drawn 
from  the  observed  event. 

Here  is  presented  the  question  raised  by  some 
philosophers  touching  the  influence  of  the  past  upon 
the  probability  of  the  future.  Let  us  suppose  at  the 
play  of  heads  and  tails  that  heads  has  occurred  oftener 
than  tails.  By  this  alone  we  shall  be  led  to  believe 
that  in  the  constitution  of  the  coin  there  is  a  secret 
cause  which  favors  it.  Thus  in  the  conduct  of  life 
constant  happiness  is  a  proof  of  competency  which 
should  induce  us  to  employ  preferably  happy  persons. 
But  if  by  the  unreliability  of  circumstances  we  are  con- 
stantly brought  back  to  a  state  of  absolute  indecision, 
if,  for  example,  we  change  the  coin  at  each  throw  at  the 
play  of  heads  and  tails,  the  past  can  shed  no  light  upon 
the  future  and  it  would  be  absurd  to  take  account  of  it. 

Sixth  Principle. — Each  of  the  causes  to  which  an 
observed  event  may  be  attributed  is  indicated  with  just 
as  much  likelihood  as  there  is  probability  that  the  event 
will  take  place,  supposing  the  event  to  be  constant. 
The  probability  of  the  existence  of  any  one  of  these 
causes  is  then  a  fraction  whose  numerator  is  the  prob- 
ability of  the  event  resulting  from  this  cause  and  whose 
denominator  is  the  sum  of  the  similar  probabilities 
relative  to  all  the  causes;  if  these  various  causes,  con- 
sidered a  priori,  are  unequally  probable,  it  is  necessary, 


1 6        A  PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 

in  place  of  the  probability  of  the  event  resulting  from 
each  cause,  to  employ  the  product  of  this  probability 
by  the  possibility  of  the  cause  itself.  This  is  the  funda- 
mental principle  of  this  branch  of  the  analysis  of  chances 
which  consists  in  passing  from  events  to  causes. 

This  principle  gives  the  reason  why  we  attribute 
regular  events  to  a  particular  cause.  Some  philosophers 
have  thought  that  these  events  are  less  possible  than 
others  and  that  at  the  play  of  heads  and  tails,  for 
example,  the  combination  in  which  heads  occurs  twenty 
successive  times  is  less  easy  in  its  nature  than  those 
where  heads  and  tails  are  mixed  in  an  irregular  manner. 
But  this  opinion  supposes  that  past  events  have  an 
influence  on  the  possibility  of  future  events,  which  is 
not  at  all  admissible.  The  regular  combinations  occur 
more  rarely  only  because  they  are  less  numerous.  If 
we  seek  a  cause  wherever  we  perceive  symmetry,  it  is 
not  that  we  regard  a  symmetrical  event  as  less  possible 
than  the  others,  but,  since  this  event  ought  to  be  the 
effect  of  a  regular  cause  or  that  of  chance,  the  first  of 
these  suppositions  is  more  probable  than  the  second. 
On  a  table  we  see  letters  arranged  in  this  order, 
Constantinople,  and  we  judge  that  this  arrange- 
ment is  not  the  result  of  chance,  not  because  it  is  less 
possible  than  the  others,  for  if  this  word  were  not 
employed  in  any  language  we  should  not  suspect  it 
came  from  any  particular  cause,  but  this  word  being  in 
use  among  us,  it  is  incomparably  more  probable  that 
some  person  has  thus  arranged  the  aforesaid  letters 
than  that  this  arrangement  is  due  to  chance. 

This  is  the  place  to  define  the  word  extraordinary. 
We  arrange  in  our  thought  all  possible  events  in  various 


CALCULUS    OF  PROBABILITIES.  1? 

classes ;  and  we  regard  as  extraordinary  those  classes 
which  include  a  very  small  number.  Thus  at  the  play 
of  heads  and  tails  the  occurrence  of  heads  a  hundred 
successive  times  appears  to  us  extraordinary  because  of 
the  almost  infinite  number  of  combinations  which  may 
occur  in  a  hundred  throws;  and  if  we  divide  the  com- 
binations into  regular  series  containing  an  order  easy 
to  comprehend,  and  into  irregular  series,  the  latter  are 
incomparably  more  numerous.  The  drawing  of  a 
white  ball  from  an  urn  which  among  a  million  balls 
contains  only  one  of  this  color,  the  others  being  black, 
would  appear  to  us  likewise  extraordinary,  because  we 
form  only  two  classes  of  events  relative  to  the  two 
colors.  But  the  drawing  of  the  number  475813,  for 
example,  from  an  urn  that  contains  a  million  numbers 
seems  to  us  an  ordinary  event;  because,  comparing 
individually  the  numbers  with  one  another  without 
dividing  them  into  classes,  we  have  no  reason  to 
believe  that  one  of  them  will  appear  sooner  than  the 
others. 

From  what  precedes,  we  ought  generally  to  conclude 
that  the  more  extraordinary  the  event,  the  greater  the 
need  of  its  being  supported  by  strong  proofs.  For 
those  who  attest  it,  being  able  to  deceive  or  to  have 
been  deceived,  these  two  causes  are  as  much  more 
probable  as  the  reality  of  the  event  is  less.  We  shall 
see  this  particularly  when  we  come  to  speak  of  the 
probability  of  testimony. 

Seventh  Principle.  — The  probability  of  a  future  event 
is  the  sum  of  the  products  of  the  probability  of  each 
cause,  drawn  from  the  event  observed,  by  the  prob- 
ability that,  this  cause  existing,  the  future  event  will 


1 8        A   PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

occur.  The  following  example  will  illustrate  this 
principle. 

Let  us  imagine  an  urn  which  contains  only  two  balls, 
each  of  which  may  be  either  white  or  black.  One  of 
these  balls  is  drawn  and  is  put  back  into  the  urn  before 
proceeding  to  a  new  draw.  Suppose  that  in  the  first 
two  draws  white  balls  have  been  drawn;  the  prob- 
ability of  again  drawing  a  white  ball  at  the  third  draw 
is  required. 

Only  two  hypotheses  can  be  made  here :  either  one 
of  the  balls  is  white  and  the  other  black,  or  both  are 
white.  In  the  first  hypothesis  the  probability  of  the 
event  observed  is  J;  it  is  unity  or  certainty  in  the 
second.  Thus  in  regarding  these  hypotheses  as  so 
many  causes,  we  shall  have  for  the  sixth  principle 
%  and  |  for  their  respective  probabilities.  But  if  the 
first  hypothesis  occurs,  the  probability  of  drawing  a 
white  ball  at  the  third  draw  is  ^ ;  it  is  equal  to  certainty 
in  the  second  hypothesis ;  multiplying  then  the  last 
probabilities  by  those  of  the  corresponding  hypotheses, 
the  sum  of  the  products,  or  T9^,  will  be  the  probability 
of  drawing  a  white  ball  at  the  third  draw. 

When  the  probability  of  a  single  event  is  unknown 
we  may  suppose  it  equal  to  any  value  from  zero  to 
unity.  The  probability  of  each  of  these  hypotheses, 
drawn  from  the  event  observed,  is,  by  the  sixth  prin- 
ciple, a  fraction  whose  numerator  is  the  probability  of 
the  event  in  this  hypothesis  and  whose  denominator  is 
the  sum  of  the  similar  probabilities  relative  to  all  the 
hypotheses.  Thus  the  probability  that  the  possibility 
of  the  event  is  comprised  within  given  limits  is  the  sum 
of  the  fractions  comprised  within  these  limits.  Now  if 


CALCULUS  OF  PROBABILITIES.  19 

we  multiply  each  fraction  by  the  probability  of  the 
future  event,  determined  in  the  corresponding  hypothe- 
sis, the  sum  of  the  products  relative  to  all  the  hypotheses 
will  be,  by  the  seventh  principle,  the  probability  of  the 
future  event  drawn  from  the  event  observed.  Thus 
we  find  that  an  event  having  occurred  successively  any 
number  of  times,  the  probability  that  it  will  happen 
again  the  next  time  is  equal  to  this  number  increased 
by  unity  divided  by  the  same  number,  increased  by 
two  units.  Placing  the  most  ancient  epoch  of  history 
at  five  thousand  years  ago,  or  at  182623  days,  and  the 
sun  having  risen  constantly  in  the  interval  at  each 
revolution  of  twenty-four  hours,  it  is  a  bet  of  1826214 
to  one  that  it  will  rise  again  to-morrow.  But  this 
number  is  incomparably  greater  for  him  who,  recogniz- 
ing in  the  totality  of  phenomena  the  principal  regulator 
of  days  and  seasons,  sees  that  nothing  at  the  present 
moment  can  arrest  the  course  of  it. 

Buffon  in  his  Political  Arithmetic  calculates  differently 
the  preceding  probability.  He  supposes  that  it  differs 
from  unity  only  by  a  fraction  whose  numerator  is  unity 
and  whose  denominator  is  the  number  2  raised  to  a 
power  equal  to  the  number  of  days  which  have  elapsed 
since  the  epoch.  But  the  true  manner  of  relating 
past  events  with  the  probability  of  causes  and  of  future 
events  was  unknown  to  this  illustrious  writer. 


CHAPTER   IV. 
CONCERNING  HOPE. 

THE  probability  of  events  serves  to  determine  the 
hope  or  the  fear  of  persons  interested  in  their  exist- 
ence. The  word  hope  has  various  acceptations;  it 
expresses  generally  the  advantage  of  that  one  who 
expects  a  certain  benefit  in  suppositions  which  are  only 
probable.  This  advantage  in  the  theory  of  chance  is 
a  product  of  the  sum  hoped  for  by  the  probability  of 
obtaining  it;  it  is  the  partial  sum  which  ought  to  result 
when  we  do  not  wish  to  run  the  risks  of  the  event  in 
supposing  that  the  division  is  made  proportional  to  the 
probabilities.  This  division  is  the  only  equitable  one 
when  all  strange  circumstances  are  eliminated;  because 
an  equal  degree  of  probability  gives  an  equal  right  to 
the  sum  hoped  for.  We  will  call  this  advantage 
mathematical  hope. 

Eighth  Principle. — When  the  advantage  depends  on 
several  events  it  is  obtained  by  taking  the  sum  of  the 
products  of  the  probability  of  each  event  by  the  benefit 
attached  to  its  occurrence. 

Let  us  apply  this  principle  to  some  examples.      Let 


CONCERNING  HOPE.  21 

us  suppose  that  at  the  play  of  heads  and  tails  Paul 
receives  two  francs  if  he  throws  heads  at  the  first  throw 
and  five  francs  if  he  throws  it  only  at  the  second. 
Multiplying  two  francs  by  the  probability  £  of  the  first 
case,  and  five  francs  by  the  probability  £  of  the  second 
case,  the  sum  of  the  products,  or  two  and  a  quarter 
francs,  will  be  Paul's  advantage.  It  is  the  sum  which 
he  ought  to  give  in  advance  to  that  one  who  has  given 
him  this  advantage;  for,  in  order  to  maintain  the 
equality  of  the  play,  the  throw  ought  to  be  equal  to 
the  advantage  which  it  procures. 

If  Paul  receives  two  francs  by  throwing  heads  at  the 
first  and  five  francs  by  throwing  it  at  the  second  throw, 
whether  he  has  thrown  it  or  not  at  the  first,  the  prob- 
ability of  throwing  heads  at  the  second  throw  being  £, 
multiplying  two  francs  and  five  francs  by  £  the  sum  of 
these  products  will  give  three  and  one  half  francs  for 
Paul's  advantage  and  consequently  for  his  stake  at  the 
game. 

Ninth  Principle. — In  a  series  of  probable  events  of 
which  the  ones  produce  a  benefit  and  the  others  a  loss, 
we  shall  have  the  advantage  which  results  from  it  by 
making  a  sum  of  the  products  of  the  probability  of  each 
favorable  event  by  the  benefit  which  it  procures,  and 
subtracting  from  this  sum  that  of  the  products  of  the 
probability  of  each  unfavorable  event  by  the  loss  which 
is  attached  to  it.  If  the  second  sum  is  greater  than  the 
first,  the  benefit  becomes  a  loss  and  hope  is  changed  to 
fear. 

Consequently  we  ought  always  in  the  conduct  of  life 
to  make  the  product  of  the  benefit  hoped  for,  by  its 
probability,  at  least  equal  to  the  similar  product  relative 


22        A  PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

to  the  loss.  But  it  is  necessary,  in  order  to  attain  this, 
to  appreciate  exactly  the  advantages,  the  losses,  and 
their  respective  probabilities.  For  this  a  great  accuracy 
of  mind,  a  delicate  judgment,  and  a  great  experience 
in  affairs  is  necessary ;  it  is  necessary  to  know  how  to 
guard  one's  self  against  prejudices,  illusions  of  fear  or 
hope,  and  erroneous  ideas,  ideas  of  fortune  and  happi- 
ness, with  which  the  majority  of  people  feed  their  self- 
love. 

The  application  of  the  preceding  principles  to  the 
following  question  has  greatly  exercised  the  geometri- 
cians. Paul  plays  at  heads  and  tails  with  the  condition 
of  receiving  two  francs  if  he  throws  heads  at  the  first 
thro\v,  four  francs  if  he  throws  it  only  at  the  second 
throw,  eight  francs  if  he  throws  it  only  at  the  third, 
and  so  on.  His  stake  at  the  play  ought  to  be,  accord- 
ing to  the  eighth  principle,  equal  to  the  number  of 
throws,  so  that  if  the  game  continues  to  infinity  the 
stake  ought  to  be  infinite.  However,  no  reasonable 
man  would  wish  to  risk  at  this  game  even  a  small  sum, 
for  example  five  francs.  Whence  comes  this  differ- 
ence between  the  result  of  calculation  and  the  indication 
of  common  sense  ?  We  soon  recognize  that  it  amounts 
to  this :  that  the  moral  advantage  which  a  benefit  pro- 
cures for  us  is  not  proportional  to  this  benefit  and  that 
it  depends  upon  a  thousand  circumstances,  often  very 
difficult  to  define,  but  of  which  the  most  general  and 
most  important  is  that  of  fortune. 

Indeed  it  is  apparent  that  one  franc  has  much  greater 
value  for  him  who  possesses  only  a  hundred  than  for  a 
millionaire.  We  ought  then  to  distinguish  in  the 
hoped-for  benefit  its  absolute  from  its  relative  value. 


CONCERNING  HOPE.  23 

But  the  latter  is  regulated  by  the  motives  which  make 
it  desirable,  whereas  the  first  is  independent  of  them. 
The  general  principle  for  appreciating  this  relative 
value  cannot  be  given,  but  here  is  one  proposed  by 
Daniel  Bernoulli  which  will  serve  in  many  cases. 

Tenth  Principle. — The  relative  value  of  an  infinitely 
small  sum  is  equal  to  its  absolute  value  divided  by  the 
total  benefit  of  the  person  interested.  This  supposes 
that  every  one  has  a  certain  benefit  whose  value  can 
never  be  estimated  as  zero.  Indeed  even  that  one  who 
possesses  nothing  always  gives  to  the  product  of  his 
labor  and  to  his  hopes  a  value  at  least  equal  to  that 
which  is  absolutely  necessary  to  sustain  him. 

If  we  apply  analysis  to  the  principle  just  propounded, 
we  obtain  the  following  rule :  Let  us  designate  by  unity 
the  part  of  the  fortune  of  an  individual,  independent  of 
his  expectations.  If  we  determine  the  different  values 
that  this  fortune  may  have  by  virtue  of  these  expecta- 
tions and  their  probabilities,  the  product  of  these  values 
raised  respectively  to  the  powers  indicated  by  their 
probabilities  will  be  the  physical  fortune  which  would 
procure  for  the  individual  the  same  moral  advantage 
which  he  receives  from  the  part  of  his  fortune  taken  as 
unity  and  from  his  expectations ;  by  subtracting  unity 
from  the  product,  the  difference  will  be  the  increase  of 
the  physical  fortune  due  to  expectations :  we  will  call 
this  increase  moral  hope.  It  is  easy  to  see  that  it  coin- 
cides with  mathematical  hope  when  the  fortune  taken 
as  unity  becomes  infinite  in  reference  to  the  variations 
which  it  receives  from  the  expectations.  But  when 
these  variations  are  an  appreciable  part  of  this  unity 


24        A   PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

the  two  hopes  may  differ  very  materially  among  them- 
selves. 

This  rule  conduces  to  results  conformable  to  the 
indications  of  common  sense  which  can  by  this  means 
be  appreciated  with  some  exactitude.  Thus  in  the 
preceding  question  it  is  found  that  if  the  fortune  of 
Paul  is  two  hundred  francs,  he  ought  not  reasonably  to 
stake  more  than  nine  francs.  The  same  rule  leads  us 
again  to  distribute  the  danger  over  several  parts  of  a 
benefit  expected  rather  than  to  expose  the  entire  benefit 
to  this  danger.  It  results  similarly  that  at  the  fairest 
game  the  loss  is  always  greater  than  the  gain.  Let 
us  suppose,  for  example,  that  a  player  having  a  fortune 
of  one  hundred  francs  risks  fifty  at  the  play  of  heads  and 
tails;  his  fortune  after  his  stake  at  the  play  will  be 
reduced  to  eighty-seven  francs,  that  is  to  say,  this  last 
sum  would  procure  for  the  player  the  same  moral 
advantage  as  the  state  of  his  fortune  after  the  stake. 
The  play  is  then  disadvantageous  even  in  the  case 
where  the  stake  is  equal  to  the  product  of  the  sum 
hoped  for,  by  its  probability.  We  can  judge  by  this 
of  the  immorality  of  games  in  which  the  sum  hoped  for 
is  below  this  product.  They  subsist  only  by  false 
reasonings  and  by  the  cupidity  which  they  excite  and 
which,  leading  the  people  to  sacrifice  their  necessaries 
to  chimerical  hopes  whose  improbability  they  are  not 
in  condition  to  appreciate,  are  the  source  of  an  infinity 
of  evils. 

The  disadvantage  of  games  of  chance,  the  advantage 
of  not  exposing  to  the  same  danger  the  whole  benefit 
that  is  expected,  and  all  the  similar  results  indicated  by 
common  sense,  subsist,  whatever  may  be  the  function 


CONCERNING  HOPE.  25 

of  the  physical  fortune  which  for  each  individual 
expresses  his  moral  fortune.  It  is  enough  that  the 
proportion  of  the  increase  of  this  function  to  the 
increase  of  the  physical  fortune  diminishes  in  the 
measure  that  the  latter  increases. 


CHAPTER   V. 

CONCERNING    THE   ANALYTICAL   METHODS    OF 
THE  CALCULUS  OF  PROBABILITIES. 

THE  application  of  the  principle  which  we  have  just 
expounded  to  the  various  questions  of  probability 
requires  methods  whose  investigation  has  given  birth 
to  several  methods  of  analysis  and  especially  to  the 
theory  of  combinations  and  to  the  calculus  of  finite 
differences. 

If  we  form  the  product  of  the  binomials,  unity  plus 
the  first  letter,  unity  plus  the  second  letter,  unity  plus 
the  third  letter,  and  so  on  up  to  n  letters,  and  sub- 
tract unity  from  this  developed  product,  the  result 
will  be  the  sum  of  the  combination  of  all  these  letters 
taken  one  by  one,  two  by  two,  three  by  three,  etc., 
each  combination  having  unity  for  a  coefficient.  In 
order  to  have  the  number  of  combinations  of  these  n 
letters  taken  s  by  s  times,  we  shall  observe  that  if  we 
suppose  these  letters  equal  among  themselves,  the  pre- 
ceding product  will  become  the  nth  power  of  the 
binomial  one  plus  the  first  letter;  thus  the  number  of 
combinations  of  n  letters  taken  s  by  s  times  will  be  the 
coefficient  of  the  sth  power  of  the  first  letter  in  the 


THE   CALCULUS   OF  PROBABILITIES.  27 

development  in  this  binomial ;  and  this  number  is 
obtained  by  means  of  the  known  binomial  formula. 

Attention  must  be  paid  to  the  respective  situations 
of  the  letters  in  each  combination,  observing  that  if  a 
second  letter  is  joined  to  the  first  it  may  be  placed  in 
the  first  or  second  position  which  gives  two  combina- 
tions. If  we  join  to  these  combinations  a  third  letter, 
we  can  give  it  in  each  combination  the  first,  the  second, 
and  the  third  rank  which  forms  three  combinations 
relative  to  each  of  the  two  others,  in  all  six  combina- 
tions. From  this  it  is  easy  to  conclude  that  the 
number  of  arrangements  of  which  s  letters  are  suscepti- 
ble is  the  product  of  the  numbers  from  unity  to  s.  In 
order  to  pay  regard  to  the  respective  positions  of  the 
letters  it  is  necessary  then  to  multiply  by  this  product 
the  number  of  combinations  of  n  letters  s  by  s  times, 
which  is  tantamount  to  taking  away  the  denominator 
of  the  coefficient  of  the  binomial  which  expresses  this 
number. 

Let  us  imagine  a  lottery  composed  of  n  numbers,  of 
which  r  are  drawn  at  each  draw.  The  probability  is 
demanded  of  the  drawing  of  s  given  numbers  in  one 
draw.  To  arrive  at  this  let  us  form  a  fraction  whose 
denominator  will  be  the  number  of  all  the  cases  possi- 
ble or  of  the  combinations  of  n  letters  taken  r  by  r 
times,  and  whose  numerator  will  be  the  number  of  all 
the  combinations  which  contain  the  given  s  numbers. 
This  last  number  is  evidently  that  of  the  combinations 
of  the  other  numbers  taken  n  less  s  by  n  less  s  times. 
This  fraction  will  be  the  required  probability,  and  we 
shall  easily  find  that  it  can  be  reduced  to  a  fraction 
whose  numerator  is  the  number  of  combinations  of  r 


28        A  PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 

numbers  taken  s  by  s  times,  and  whose  denominator  is 
the  number  of  combinations  of  n  numbers  taken 
similarly  s  by  s  times.  Thus  in  the  lottery  of  France, 
formed  as  is  known  of  90  numbers  of  which  five  are 
drawn  at  each  draw,  the  probability  of  drawing  a  given 
combination  is  -&>  or  TV ;  the  lottery  ought  then  for  the 
equality  of  the  play  to  give  eighteen  times  the  stake. 
The  total  number  of  combinations  two  by  two  of  the 
90  numbers  is  4005 ,  and  that  of  the  combinations  two 
by  two  of  5  numbers  is  10.  The  probability  of  the 
drawing  of  a  given  pair  is  then  3-^-5-,  and  the  lottery 
ought  to  give  four  hundred  and  a  half  times  the  stake ; 
it  ought  to  give  11748  times  for  a  given  tray,  511038 
times  for  a  quaternary,  and  43949268  times  for  a  quint. 
The  lottery  is  far  from  giving  the  player  these  advan- 
tages. 

Suppose  in  an  urn  a  white  balls,  b  black  balls,  and 
after  having  drawn  a  ball  it  is  put  back  into  the  urn ; 
the  probability  is  asked  that  in  «  number  of  draws  m 
white  balls  and  n  —  m  black  balls  will  be  drawn.  It 
is  clear  that  the  number  of  cases  that  may  occur  at 
each  drawing  is  a  -j-  b.  Each  case  of  the  second 
drawing  being  able  to  combine  with  all  the  cases  of  the 
first,  the  number  of  possible  cases  in  two  drawings  is 
the  square  of  the  binomial  a-\-b.  In  the  development 
of  this  square,  the  square  of  a  expresses  the  number  of 
cases  in  which  a  white  ball  is  twice  drawn,  the  double 
product  of  a  by  b  expresses  the  number  of  cases  in 
which  a  white  ball  and  a  black  ball  are  drawn.  Finally, 
the  square  of  b  expresses  the  number  of  cases  in  which 
two  black  balls  are  drawn.  Continuing  thus,  we  see 
generally  that  the  «th  power  of  the  binomial  a  +  b 


THE   CALCULUS  OF  PROBABILITIES.  29 

expresses  the  number  of  all  the  cases  possible  in  n 
draws;  and  that  in  the  development  of  this  power  the 
term  multiplied  by  the  mth  power  of  a  expresses  the 
number  of  cases  in  which  m  white  balls  and  n  —  in 
black  balls  may  be  drawn.  Dividing  then  this  term 
by  the  entire  power  of  the  binomial,  we  shall  have  the 
probability  of  drawing  m  white  balls  and  n  —  m  black 
balls.  The  ratio  of  the  numbers  a  and  a  -\-  b  being 
the  probability  of  drawing  one  white  ball  at  one  draw; 
and  the  ratio  of  the  numbers  b  and  a  -\-  b  being  the 
probability  of  drawing  one  black  ball ;  if  we  call  these 
probabilities/  and  g,  the  probability  of  drawing  m  white 
balls  in  n  draws  will  be  the  term  multiplied  by  the  mth 
power  of/  in  the  development  of  the  «th  power  of  the 
binomial  P  -\-  q\  we  may  see  that  the  sum  p  -)-  q  is 
unity.  This  remarkable  property  of  the  binomial  is 
very  useful  in  the  theory  of  probabilities.  But  the 
most  general  and  direct  method  of  resolving  questions 
of  probability  consists  in  making  them  depend  upon 
equations  of  differences.  Comparing  the  successive 
conditions  of  the  function  which  expresses  the  prob- 
ability when  we  increase  the  variables  by  their  respect- 
ive differences,  the  proposed  question  often  furnishes  a 
very  simple  proportion  between  the  conditions.  This 
proportion  is  what  is  called  equation  of  ordinary  or 
partial  differentials;  ordinary  when  there  is  only  one 
variable,  partial  when  there  are  several.  Let  us  con- 
sider some  examples  of  this. 

Three  players  of  supposed  equal  ability  play  together 
on  the  following  conditions :  that  one  of  the  first  two 
players  who  beats  his  adversary  plays  the  third,  and  if 
he  beats  him  the  game  is  finished.  If  he  is  beaten,  the 


30        A   PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

victor  plays  against  the  second  until  one  of  the  players 
has  defeated  consecutively  the  two  others,  which  ends 
the  game.  The  probability  is  demanded  that  the  game 
will  be  finished  in  a  certain  number  n  of  plays.  Let 
us  find  the  probability  that  it  will  end  precisely  at  the 
nth  play.  For  that  the  player  who  wins  ought  to  enter 
the  game  at  the  play  n  —  I  and  win  it  thus  at  the  fol- 
lowing play.  But  if  in  place  of  winning  the  play  n  —  i 
he  should  be  beaten  by  his  adversary  who  had  just 
beaten  the  other  player,  the  game  would  end  at  this 
play.  Thus  the  probability  that  one  of  the  players  will 
enter  the  game  at  the  play  «  —  I  and  will  win  it  is 
equal  to  the  probability  that  the  game  will  end  pre- 
cisely with  this  play;  and  as  this  player  ought  to  win 
the  following  play  in  order  that  the  game  may  be 
finished  at  the  nth  play,  the  probability  of  this  last  case 
will  be  only  one  half  of  the  preceding  one.  This 
probability  is  evidently  a  function  of  the  number  «;  this 
function  is  then  equal  to  the  half  of  the  same  function 
when  n  is  diminished  by  unity.  This  equality  forms 
one  of  those  equations  called  ordinary  finite  differential 
equations. 

We  may  easily  determine  by  its  use  the  probability 
that  the  game  will  end  precisely  at  a  certain  play.  It 
is  evident  that  the  play  cannot  end  sooner  than  at  the 
second  play;  and  for  this  it  is  necessary  that  that  one 
of  the  first  two  players  who  has  beaten  his  adversary 
should  beat  at  the  second  play  the  third  player;  the 
probability  that  the  game  will  end  at  this  play  is  £. 
Hence  by  virtue  of  the  preceding  equation  we  conclude 
that  the  successive  probabilities  of  the  end  of  the  game 
are  £  for  the  third  play,  \  for  the  fourth  play,  and  so 


THE   CALCULUS   OF  PROBABILITIES.  31 

on ;  and  in  general  £  raised  to  the  power  n  —  I  for  the 
nth  play.  The  sum  of  all  these  powers  of  £  is  unity 
less  the  last  of  these  powers ;  it  is  the  probability  that 
the  game  will  end  at  the  latest  in  n  plays. 

Let  us  consider  again  the  first  problem  more  difficult 
which  may  be  solved  by  probabilities  and  which  Pascal 
proposed  to  Fermat  to  solve.  Two  players,  A  and  B, 
of  equal  skill  play  together  on  the  conditions  that  the 
one  who  first  shall  beat  the  other  a  given  number  of 
times  shall  win  the  game  and  shall  take  the  sum  of  the 
stakes  at  the  game;  after  some  throws  the  players 
agree  to  quit  without  having  finished  the  game :  we  ask 
in  what  manner  the  sum  ought  to  be  divided  between 
them.  It  is  evident  that  the  parts  ought  to  be  propor- 
tional to  the  respective  probabilities  of  winning  the 
game.  The  question  is  reduced  then  to  the  determina- 
tion of  these  probabilities.  They  depend  evidently 
upon  the  number  of  points  which  each  player  lacks  of 
having  attained  the  given  number.  Hence  the  prob- 
ability of  A  is  a  function  of  the  two  numbers  which  we 
will  call  indices.  If  the  two  players  should  agree  to 
play  one  throw  more  (an  agreement  which  does  not 
change  their  condition,  provided  that  after  this  new 
throw  the  division  is  always  made  proportionally  to  the 
new  probabilities  of  winning  the  game),  then  either  A 
would  win  this  throw  and  in  that  case  the  number  of 
points  which  he  lacks  would  be  diminished  by  unity, 
or  the  player  B  would  win  it  and  in  that  case  the 
number  of  points  lacking  to  this  last  player  would  be 
less  by  unity.  But  the  probability  of  each  of  these 
cases  is  \ ;  the  function  sought  is  then  equal  to  one  half 
of  this  function  in  which  we  diminish  by  unity  the  first 


32        A  PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 

index  plus  the  half  of  the  same  function  in  which  the 
second  variable  is  diminished  by  unity.  This  equality 
is  one  of  those  equations  called  equations  of  partial 
differentials. 

We  are  able  to  determine  by  its  use  the  probabilities 
of  A  by  dividing  the  smallest  numbers,  and  by  observ- 
ing that  the  probability  or  the  function  which  expresses 
it  is  equal  to  unity  when  the  player  A  does  not  lack  a 
single  point,  or  when  the  first  index  is  zero,  and  that 
this  function  becomes  zero  with  the  second  index.  Sup- 
posing thus  that  the  player  A  lacks  only  one  point,  we 
find  that  his  probability  is  f,  f,  |,  etc.,  according  as  B 
lacks  one  point,  two,  three,  etc.  Generally  it  is  then 
unity  less  the  power  of  £,  equal  to  the  number  of  points 
which  B  lacks.  We  will  suppose  then  that  the  player 
A  lacks  two  points  and  his  probability  will  be  found 
equal  to  J,  £,  \\,  etc.,  according  as  B  lacks  one  point, 
two  points,  three  points,  etc.  We  will  suppose  again 
that  the  player  A  lacks  three  points,  and  so  on. 

This  manner  of  obtaining  the  successive  values  of  a 
quantity  by  means  of  its  equation  of  differences  is  long 
and  laborious .  The  geometricians  have  sought  methods 
to  obtain  the  general  function  of  indices  that  satisfies 
this  equation,  so  that  for  any  particular  case  we  need 
only  to  substitute  in  this  function  the  corresponding 
values  of  the  indices.  Let  us  consider  this  subject  in 
a  general  way.  For  this  purpose  'let  us  conceive  a 
series  of  terms  arranged  along  a  horizontal  line  so  that 
each  of  them  is  derived  from  the  preceding  one  accord- 
ing to  a  given  law.  Let  us  suppose  this  law  expressed 
by  an  equation  among  several  consecutive  terms  and 
their  index,  or  the  number  which  indicates  the  rank  that 


THE  CALCULUS  OF  PROBABILITIES.  33 

they  occupy  in  the  series.  This  equation  I  call  the 
equation  of  finite  differences  by  a  single  index.  The 
order  or  the  degree  of  this  equation  is  the  difference  of 
rank  of  its  two  extreme  terms.  We  are  able  by  its  use 
to  determine  successively  the  terms  of  the  series  and  to 
continue  it  indefinitely ;  but  for  that  it  is  necessary  to 
know  a  number  of  terms  of  the  series  equal  to  the 
degree  of  the  equation .  These  terms  are  the  arbitrary 
constants  of  the  expression  of  the  general  term  of  the 
series  or  of  the  integral  of  the  equation  of  differences. 

Let  us  imagine  now  below  the  terms  of  the  preceding 
series  a  second  series  of  terms  arranged  horizontally; 
let  us  imagine  again  below  the  terms  of  the  second 
series  a  third  horizontal  series,  and  so  on  to  infinity; 
and  let  us  suppose  the  terms  of  all  these  series  con- 
nected by  a  general  equation  among  several  consecutive 
terms,  taken  as  much  in  the  horizontal  as  in  the  ver- 
tical sense,  and  the  numbers  which  indicate  their  rank 
in  the  two  senses.  This  equation  is  called  the  equation 
of  partial  finite  differences  by  two  indices. 

Let  us  imagine  in  the  same  way  below  the  plan  of 
the  preceding  series  a  second  plan  of  similar  series, 
whose  terms  should  be  placed  respectively  below  those 
of  the  first  plan  ;  let  us  imagine  again  below  this  second 
plan  a  third  plan  of  similar  series,  and  so  on  to  infinity; 
let  us  suppose  all  the  terms  of  these  series  connected 
by  an  equation  among  several  consecutive  terms  taken 
in  the  sense  of  length,  width,  and  depth,  and  the  three 
numbers  which  indicate  their  rank  in  these  three  senses. 
This  equation  I  call  the  equation  of  partial  finite  differ- 
ences by  three  indices. 

Finally,  considering  the  matter  in  an  abstract  way 


34        A  PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

and  independently  of  the  dimensions  of  space,  let  us 
imagine  generally  a  system  of  magnitudes,  which  should 
be  functions  of  a  certain  number  of  indices,  and  let  us 
suppose  among  these  magnitudes,  their  relative  differ- 
ences to  these  indices  and  the  indices  themselves,  as 
many  equations  as  there  are  magnitudes ;  these  equa- 
tions will  be  partial  finite  differences  by  a  certain  num- 
ber of  indices. 

We  are  able  by  their  use  to  determine  successively 
these  magnitudes.  But  in  the  same  manner  as  the 
equation  by  a  single  index  requires  for  it  that  we 
known  a  certain  number  of  terms  of  the  series,  so  the 
equation  by  two  indices  requires  that  we  know  one  or 
several  lines  of  series  whose  general  terms  should  be 
expressed  each  by  an  arbitrary  function  of  one  of  the 
indices.  Similarly  the  equation  by  three  indices 
requires  that  we  know  one  or  several  plans  of  series, 
the  general  terms  of  which  should  be  expressed  each 
by  an  arbitrary  function  of  two  indices,  and  so  on.  In 
all  these  cases  we  shall  be  able  by  successive  elimina- 
tions to  determine  a  certain  term  of  the  series.  But 
all  the  equations  among  which  we  eliminate  being 
comprised  in  the  same  system  of  equations,  all  the 
expressions  of  the  successive  terms  which  we  obtain  by 
these  eliminations  ought  to  be  comprised  in  one  general 
expression,  a  function  of  the  indices  which  determine 
the  rank  of  the  term.  This  expression  is  the  integral 
of  the  proposed  equation  of  differences,  and  the  search 
for  it  is  the  object  of  integral  calculus. 

Taylor  is  the  first  who  in  his  work  entitled  Mctodus 
incrementorum  has  considered  linear  equations  of  finite 
differences.  He  gives  the  manner  of  integrating  those 


THE  CALCULUS   OF  PROBABILITIES.  35 

of  the  first  order  with  a  coefficient  and  a  last  term, 
functions  of  the  index.  In  truth  the  relations  of  the 
terms  of  the  arithmetical  and  geometrical  progressions 
which  have  always  been  taken  into  consideration  are 
the  simplest  cases  of  linear  equations  of  differences ;  but 
they  had  not  been  considered  from  this  point  of  view. 
It  was  one  of  those  which,  attaching  themselves  to 
general  theories,  lead  to  these  theories  and  are  conse- 
quently veritable  discoveries. 

About  the  same  time  Moivre  was  considering  under 
the  name  of  recurring  series  the  equations  of  finite 
differences  of  a  certain  order  having  a  constant  coeffi- 
cient. He  succeeded  in  integrating  them  in  a  very 
ingenious  manner.  As  it  is  always  interesting  to  follow 
the  progress  of  inventors,  I  shall  expound  the  method 
of  Moivre  by  applying  it  to  a  recurring  series  whose 
relation  among  three  consecutive  terms  is  given.  First 
he  considers  the  relation  among  the  consecutive  terms 
of  a  geometrical  progression  or  the  equation  of  two 
terms  which  expresses  it.  Referring  it  to  terms  less 
than  unity,  he  multiplies  it  in  this  state  by  a  constant 
factor  and  subtracts  the  product  from  the  first  equation. 
Thus  he  obtains  an  equation  among  three  consecutive 
terms  of  the  geometrical  progression.  Moivre  considers 
next  a  second  progression  whose  ratio  of  terms  is  the 
same  factor  which  he  has  just  used.  He  diminishes 
similarly  by  unity  the  index  of  the  terms  of  the  equa- 
tion of  this  new  progression.  In  this  condition  he 
multiplies  it  by  the  ratio  of  the  terms  of  the  first  pro- 
gression, and  he  subtracts  the  product  from  the  equation 
of  the  second  progression,  which  gives  him  among  three 
consecutive  terms  of  this  progression  a  relation  entirely 


36        A  PHILOSOPHICAL  ESSAY  CW  PROBABILITIES. 

similar  to  that  which  he  has  found  for  the  first  progres- 
sion. Then  he  observes  that  if  one  adds  term  by  term 
the  two  progressions,  the  same  ratio  exists  among  any 
three  of  these  consecutive  terms.  He  compares  the 
coefficients  of  this  ratio  to  those  of  the  relation  of  the 
terms  of  the  proposed  recurrent  series,  and  he  finds  for 
determining  the  ratios  of  the  two  geometrical  progres- 
sions an  equation  of  the  second  degree,  whose  roots  are 
these  ratios.  Thus  Moivre  decomposes  the  recurrent 
series  into  two  geometrical  progressions,  each  multi- 
plied by  an  arbitrary  constant  which  he  determines  by 
means  of  the  first  two  terms  of  the  recurrent  series. 
This  ingenious  process  is  in  fact  the  one  that  d' Alembert 
has  since  employed  for  the  integration  of  linear  equa- 
tions of  infinitely  small  differences  with  constant  coeffi- 
cients, and  Lagrange  has  transformed  into  similar 
equations  of  finite  differences. 

Finally,  I  have  considered  the  linear  equations  of 
partial  finite  differences,  first  under  the  name  of  recurro- 
recurrent  series  and  afterwards  under  their  own  name. 
The  most  general  and  simplest  manner  of  integrating 
all  these  equations  appears  to  me  that  which  I  have 
based  upon  the  consideration  of  discriminant  functions, 
the  idea  of  which  is  here  given. 

If  we  conceive  a  function  V  of  a  variable  /  developed 
according  to  the  powers  of  this  variable,  the  coefficient 
of  any  one  of  these  powers  will  be  a  function  of  the 
exponent  or  index  of  this  power,  which  index  I  shall 
call  x.  V  is  what  I  call  the  discriminant  function  of. 
this  coefficient  or  of  the  function  of  the  index. 

Now  if  we  multiply  the  series  of  the  development  of 
V  by  a  function  of  the  same  variable,  such,  for  example, 


THE  CALCULUS   OF  PROBABILITIES.  3? 

as  unity  plus  two  times  this  variable,  the  product  will 
be  a  new  discriminant  function  in  which  the  coefficient 
of  the  power  x  of  the  variable  t  will  be  equal  to  the 
coefficient  of  the  same  power  in  V  plus  twice  the 
coefficient  of  the  power  less  unity.  Thus  the  function 
of  the  index  x  in  the  product  will  be  equal  to  the  func- 
tion of  the  index  x  in  V  plus  twice  the  same  function 
in  which  the  index  is  diminished  by  unity.  This  func- 
tion of  the  index  x  is  thus  a  derivative  of  the  function 
of  the  same  index  in  the  development  of  V,  a  function 
which  I  shall  call  the  primitive  function  of  the  index. 
Let  us  designate  the  derivative  function  by  the  letter  d 
placed  before  the  primitive  function.  The  derivation 
indicated  by  this  letter  will  depend  upon  the  multiplier 
of  V,  which  we  will  call  T  and  which  we  will  suppose 
developed  like  V  by  the  ratio  to  the  powers  of  the 
variable  /.  If  we  multiply  anew  by  T  the  product  of 
V  by  T,  which  is  equivalent  to  multiplying  V  by  T2, 
we  shall  form  a  third  discriminant  function,  in  which 
the  coefficient  of  the  jrth  power  of  t  will  be  a  derivative 
similar  to  the  corresponding  coefficient  of  the  preceding 
product ;  it  may  be  expressed  by  the  same  character  8 
placed  before  the  preceding  derivative,  and  then  this 
character  will  be  written  twice  before  the  primitive 
function  of  x.  But  in  place  of  writing  it  thus  twice  we 
give  it  2  for  an  exponent. 

Continuing  thus,  we  see  generally  that  if  we  multiply 
V  by  the  «th  power  of  T,  we  shall  have  the  coefficient 
of  the  ;rth  power  of  t  in  the  product  of  V  by  the  nth 
power  of  T  by  placing  before  the  primitive  function  the 
character  6  with  n  for  an  exponent. 

Let  us  suppose,  for  example,  that  T  be  unity  divided 


38        A  PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 

by  /;  then  in  the  product  of  Fby  T  the  coefficient  of 
the  .rth  power  of  /  will  be  the  coefficient  of  the  power 
greater  by  unity  in  V\  this  coefficient  in  the  product 
of  V  by  the  nth  power  of  T  will  then  be  the  primitive 
function  in  which  x  is  augmented  by  n  units. 

Let  us  consider  now  a  new  function  Z  of  /,  developed 
like  V  and  T  according  to  the  powers  of  /;  let  us 
designate  by  the  character  A  placed  before  the  primi- 
tive function  the  coefficient  of  the  .rth  power  of  /  in  the 
product  of  V  by  Z;  this  coefficient  in  the  product  of  V 
by  the  nth  power  of  Z  will  be  expressed  by  the  char- 
acter A  affected  by  the  exponent  n  and  placed  before 
the  primitive  function  of  x. 

If,  for  example,  Z  is  equal  to  unity  divided  by  t  less 
one,  the  coefficient  of  the  xth  power  of  /  in  the  product 
of  V  by  Z  will  be  the  coefficient  of  the  x  -\-  I  power 
of  t  in  V  less  the  coefficient  of  the  xth  power.  It  will 
be  then  the  finite  difference  of  the  primitive  function  of 
the  index  x.  Then  the  character  A  indicates  a  finite 
difference  of  the  primitive  function  in  the  case  where 
the  index  varies  by  unity;  and  the  nth  power  of  this 
character  placed  before  the  primitive  function  will  indi- 
cate the  finite  nth  difference  of  this  function.  If  we 
suppose  that  T  be  unity  divided  by  /,  we  shall  have  7 
equal  to  the  binomial  Z  -j-  I .  The  product  of  V  by 
the  nth  power  of  T  will  then  be  equal  to  the  product 
of  V  by  the  nth  power  of  the  binomial  Z-\-\.  Develop- 
ing this  power  in  the  ratio  of  the  powers  of  Z,  the 
product  of  V  by  the  various  terms  of  this  development 
will  be  the  discriminant  functions  of  these  same  terms 
in  which  we  substitute  in  place  of  the  powers  of  Z  the 


THE  CALCULUS  OF  PROBABILITIES.  39 

corresponding  finite  differences  of  the  primitive  function 
of  the  index. 

Now  the  product  of  V  by  the  nth  power  of  T  is  the 
primitive  function  in  which  the  index  x  is  augmented 
by  n  units;  repassing  from  the  discriminant  functions 
to  their  coefficients,  we  shall  have  this  primitive  function 
thus  augmented  equal  to  the  development  of  the  nth 
power  of  the  binomial  Z-\-  \,  provided  that  in  this 
development  we  substitute  in  place  of  the  powers  of  Z 
the  corresponding  differences  of  the  primitive  function 
and  that  we  multiply  the  independent  term  of  these 
powers  by  the  primitive  function.  We  shall  thus 
obtain  the  primitive  function  whose  index  is  augmented 
by  any  number  n  by  means  of  its  differences. 

Supposing  that  T  and  Z  always  have  the  preceding 
values,  we  shall  have  Z  equal  to  the  binomial  T —  i  ; 
the  product  of  V  by  the  #th  power  of  Z  will  then  be 
equal  to  the  product  of  V  by  the  development  of  the 
#th  power  of  the  binomial  T  —  I .  Repassing  from  the 
discriminant  functions  to  their  coefficients  as  has  just 
been  done,  we  shall  have  the  nth  difference  of  the 
primitive  function  expressed  by  the  development  of  the 
?zth  power  of  the  binomial  T  —  I ,  in  which  we  substi- 
tute for  the  powers  of  T  this  same  function  whose  index 
is  augmented  by  the  exponent  of  the  power,  and  for 
the  independent  term  of  t,  which  is  unity,  the  primitive 
function,  which  gives  this  difference  by  means  of  the 
consecutive  terms  of  this  function. 

Placing  S  before  the  primitive  function  expressing  the 
derivative  of  this  function,  which  multiplies  the  x  power 
of  /  in  the  product  of  V  by  T,  and  A  expressing  the 
same  derivative  in  the  product  of  V  by  Z,  we  are  led 


40        A  PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 

by  that  which  precedes  to  this  general  result :  whatever 
may  be  the  function  of  the  variable  /  represented  by  T 
and  Z,  we  may,  in  the  development  of  all  the  identical 
equations  susceptible  of  being  formed  among  these 
functions,  substitute  the  characters  d  and  //  in  place  of 
T  and  Z,  provided  that  we  write  the  primitive  function 
of  the  index  in  series  with  the  powers  and  with  the 
products  of  the  powers  of  the  characters,  and  that  we 
multiply  by  this  function  the  independent  terms  of  these 
characters. 

We  are  able  by  means  of  this  general  result  to  trans- 
form any  certain  power  of  a  difference  of  the  primitive 
function  of  the  index  x,  in  which  x  varies  by  unity,  into 
a  series  of  differences  of  the  same  function  in  which  x 
varies  by  a  certain  number  of  units  and  reciprocally. 
Let  us  suppose  that  T  be  the  i  power  of  unity  divided 
by  /  —  i ,  and  that  Z  be  always  unity  divided  by  /  —  I ; 
then  the  coefficient  of  the  x  power  of  /  in  the  pro- 
duct of  V  by  T  will  be  the  coefficient  of  the  x  -\-  i 
power  of  /  in  V  less  the  coefficient  of  the  x  power  of  t\ 
it  will  then  be  the  finite  difference  of  the  primitive 
function  of  the  index  x  in  which  we  vary  this  index  by 
the  number  i.  It  is  easy  to  see  that  T  is  equal  to  the 
difference  between  the  i  power  of  the  binomial  Z-f-  I 
and  unity.  The  wth  power  of  T  is  equal  to  the  «th 
power  of  this  difference.  If  in  this  equality  we  substi- 
tute in  place  of  T  and  Z  the  characters  6  and  J,  and 
after  the  development  we  place  at  the  end  of  each  term 
the  primitive  function  of  the  index  x>  we  shall  have  the 
wth  difference  of  this  function  in  which  x  varies  by  * 
units  expressed  by  a  series  of  differences  of  the  same 
function  in  which  x  varies  by  unity.  This  series  is 


THE   CALCULUS  OF  PROBABILITIES.  41 

only  a  transformation  of  the  difference  which  it 
expresses  and  which  is  identical  with  it;  but  it  is  in 
similar  transformations  that  the  power  of  analysis 
resides. 

The  generality  of  analysis  permits  us  to  suppose  in 
this  expression  that  n  is  negative.  Then  the  negative 
powers  of  tf  and  A  indicate  the  integrals.  Indeed  the 
nth  difference  of  the  primitive  function  having  for  a 
discriminant  function  the  product  of  V  by  the  nth  power 
of  the  binomial  one  divided  by  t  less  unity,  the  primi- 
tive function  which  is  the  nth  integral  of  this  difference 
has  for  a  discriminant  function  that  of  the  same  differ- 
ence multiplied  by  the  nth  power  taken  less  than  the 
binomial  one  divided  by  /  minus  one,  a  power  to  which 
the  same  power  of  the  character  A  corresponds ;  this 
power  indicates  then  an  integral  of  the  same  order,  the 
index  x  varying  by  unity;  and  the  negative  powers  of 
6  indicate  equally  the  integrals  x  varying  by  i  units. 
We  see,  thus,  in  the  clearest  and  simplest  manner  the 
rationality  of  the  analysis  observed  among  the  positive 
powers  and  differences,  and  among  the  negative  powers 
and  the  integrals. 

If  the  function  indicated  by  $  placed  before  the 
primitive  function  is  zero,  we  shall  have  an  equation  of 
finite  differences,  and  Fwill  be  the  discriminant  function 
of  its  integral.  In  order  to  obtain  this  discriminant 
function  we  shall  observe  that  in  the  product  of  V  by 
T  all  the  powers  of  /  ought  to  disappear  except  the 
powers  inferior  to  the  order  of  the  equation  of  differ- 
ences; V  is  then  equal  to  a  fraction  whose  denominator 
is  T  and  whose  numerator  is  a  polynomial  in  which  the 
highest  power  of  t  is  less  by  unity  than  the  order  of  the 


42        A  PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 

equation  of  differences.  The  arbitrary  coefficients  of 
the  various  powers  of  /  in  this  polynomial,  including 
the  power  zero,  will  be  determined  by  as  many  values 
of  the  primitive  function  of  the  index  when  we  make 
successively  x  equal  to  zero,  to  one,  to  two,  etc. 
When  the  equation  of  differences  is  given  we  determine 
T  by  putting  all  its  terms  in  the  first  member  and  zero 
in  the  second;  by  substituting  in  the  first  member  unity 
in  place  of  the  function  which  has  the  largest  index ; 
the  first  power  of  /  in  place  of  the  primitive  function  in 
which  this  index  is  diminished  by  unity;  the  second 
power  of  /  for  the  primitive  function  where  this  index 
is  diminished  by  two  units,  and  so  on.  The  coefficient 
of  the  x\\\  power  of  /  in  the  development  of  the  preced- 
ing expression  of  V  will  be  the  primitive  function  of  x 
or  the  integral  of  the  equation  of  finite  differences. 
Analysis  furnishes  for  this  development  various  means, 
among  which  we  may  choose  that  one  which  is  most 
suitable  for  the  question  proposed ;  this  is  an  advantage 
of  this  method  of  integration. 

Let  us  conceive  now  that  V  be  a  function  of  the  two 
variables  /  and  /'  developed  according  to  the  powers 
and  products  of  these  variables ;  the  coefficient  of  any 
product  of  the  powers  x  and  x'  of  /  and  /'  will  be  a 
function  of  the  exponents  or  indices  x  and  x'  of  these 
powers;  this  function  I  shall  call  the  primitive  function 
of  which  V  is  the  discriminant  function. 

Let  us  multiply  V  by  a  function  T  of  the  two 
variables  t  and  /'  developed  like  V  in  ratio  of  the 
powers  and  the  products  of  these  variables ;  the  product 
will  be  the  discriminant  function  of  a  derivative  of  the 
primitive  function;  if  T,  for  example,  is  equal  to  the 


THE   CALCULUS  OF  PROBABILITIES.  43 

variable  /  plus  the  variable  t'  minus  two,  this  derivative 
will  be  the  primitive  function  of  which  we  diminish  by 
unity  the  index  x  plus  this  same  primitive  function  of 
which  we  diminish  by  unity  the  index  x'  less  two 
times  the  primitive  function.  Designating  whatever  T 
may  be  by  the  character  d  placed  before  the  primitive 
function,  this  derivative,  the  product  of  V  by  the  wth 
power  of  T,  will  be  the  discriminant  function  of  the 
derivative  of  the  primitive  function  before  which  one 
places  the  ;/th  power  of  the  character  8.  Hence  result 
the  theorems  analogous  to  those  which  are  relative  to 
functions  of  a  single  variable. 

Suppose  the  function  indicated  by  the  character  $  be 
zero;  one  will  have  an  equation  of  partial  differences. 
If,  for  example,  we  make  as  before  T  equal  to  the 
variable  /  phis  the  variable  t'  —  2,  we  have  zero  equal 
to  the  primitive  function  of  which  we  diminish  by  unity 
the  index  x  plus  the  same  function  of  which  we  diminish 
by  unity  the  index  x'  minus  two  times  the  primitive 
function.  The  discriminant  function  V  of  the  primitive 
function  or  of  the  integral  of  this  equation  ought  then 
to  be  such  that  its  product  by  T  does  not  include  at 
all  the  products  of  /  by  t' ;  but  Fmay  include  separately 
the  powers  of  t  and  those  of  t' ,  that  is  to  say,  an  arbi- 
trary function  of  t  and  an  arbitrary  function  of  /';  V  is 
then  a  fraction  whose  numerator  is  the  sum  of  these  two 
arbitrary  functions  and  whose  denominator  is  T.  The 
coefficient  of  the  product  of  the  ;rth  power  of  t  by  the 
x'  power  of  /'  in  the  development  of  this  fraction  will 
then  be  the  integral  of  the  preceding  equation  of  partial 
differences.  This  method  of  integrating  this  kind  of 
equations  seems  to  me  the  simplest  and  the  easiest  by 


44        A  PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 

the  employment  of  the  various  analytical  processes  for 
the  development  of  rational  fractions. 

More  ample  details  in  this  matter  would  be  scarcely 
understood  without  the  aid  of  calculus. 

Considering  equations  of  infinitely  small  partial 
differences  as  equations  of  finite  partial  differences  in 
which  nothing  is  neglected,  we  are  able  to  throw  light 
upon  the  obscure  points  of  their  calculus,  which  have 
been  the  subject  of  great  discussions  among  geometri- 
cians. It  is  thus  that  I  have  demonstrated  the  possi- 
bility of  introducing  discontinued  functions  in  their 
integrals,  provided  that  the  discontinuity  takes  place 
only  for  the  differentials  of  the  order  of  these  equations 
or  of  a  superior  order.  The  transcendent  results  of 
calculus  are,  like  all  the  abstractions  of  the  understand- 
ing, general  signs  whose  true  meaning  may  be  ascer- 
tained only  by  repassing  by  metaphysical  analysis  to 
the  elementary  ideas  which  have  led  to  them ;  this 
often  presents  great  difficulties,  for  the  human  mind 
tries  still  less  to  transport  itself  into  the  future  than  to 
retire  within  itself.  The  comparison  of  infinitely  small 
differences  with  finite  differences  is  able  similarly  to 
shed  great  light  upon  the  metaphysics  of  infinitesimal 
calculus. 

It  is  easily  proven  that  the  finite  nth  difference  of  a 
function  in  which  the  increase  of  the  variable  is  E 
being  divided  by  the  nth  power  of  E,  the  quotient 
reduced  in  series  by  ratio  to  the  powers  of  the  increase 
E  is  formed  by  a  first  term  independent  of  E.  In  the 
measure  that  E  diminishes,  the  series  approaches  more 
and  more  this  first  term  from  which  it  can  differ  only 
by  quantities  less  than  any  assignable  magnitude. 


THE   CALCULUS   OF  PROBABILITIES.  45 

This  term  is  then  the  limit  of  the  series  and  expresses 
in  differential  calculus  the  infinitely  small  nth  difference 
of  the  function  divided  by  the  nth  power  of  the  infinitely 
small  increase. 

Considering  from  this  point  of  view  the  infinitely 
small  differences,  we  see  that  the  various  operations  of 
differential  calculus  amount  to  comparing  separately  in 
the  development  of  identical  expressions  the  finite 
terms  or  those  independent  of  the  increments  of  the 
variables  which  are  regarded  as  infinitely  small ;  this 
is  rigorously  exact,  these  increments  being  indetermi- 
nant.  Thus  differential  calculus  has  all  the  exactitude 
of  other  algebraic  operations. 

The  same  exactitude  is  found  in  the  applications  of 
differential  calculus  to  geometry  and  mechanics.  If 
we  imagine  a  curve  cut  by  a  secant  at  two  adjacent 
points,  naming  E  the  interval  of  the  ordinates  of  these 
two  points,  E  will  be  the  increment  of  the  abscissa  from 
the  first  to  the  second  ordinate.  It  is  easy  to  see  that 
the  corresponding  increment  of  the  ordinate  will  be  the 
product  of  E  by  the  first  ordinate  divided  by  its  sub- 
secant;  augmenting  then  in  this  equation  of  the  curve 
the  first  ordinate  by  this  increment,  we  shall  have  the 
equation  relative  to  the  second  ordinate.  The  differ- 
ence of  these  two  equations  will  be  a  third  equation 
which,  developed  by  the  ratio  of  the  powers  of  E  and 
divided  by  E,  will  have  its  first  term  independent  of  E, 
which  will  be  the  limit  of  this  development.  This 
term,  equal  to  zero,  will  give  then  the  limit  of  the  sub- 
secants,  a  limit  which  is  evidently  the  subtangent. 

This  singularly  happy  method  of  obtaining  the  sub- 
tangent  is  due  to  Fermat,  who  has  extended  it  to 


46        A  PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

transcendent  curves.  This  great  geometrician  ex- 
presses by  the  character  E  the  increment  of  the 
abscissa;  and  considering  only  the  first  power  of  this 
increment,  he  determines  exactly  as  we  do  by  differen- 
tial calculus  the  subtangents  of  the  curves,  their  points 
of  inflection,  the  maxima  and  minima  of  their  ordinates, 
and  in  general  those  of  rational  functions.  We  see 
likewise  by  his  beautiful  solution  of  the  problem  of  the 
refraction  of  light  inserted  in  the  Collection  of  the 
Letters  of  Descartes  that  he  knows  how  to  extend  his 
methods  to  irrational  functions  in  freeing  them  from 
irrationalities  by  the  elevation  of  the  roots  to  powers. 
Fermat  should  be  regarded,  then,  as  the  true  discoverer 
of  Differential  Calculus.  Newton  has  since  rendered  this 
calculus  more  analytical  in  his  Method  of  Fluxions,  and 
simplified  and  generalized  the  processes  by  his  beautiful 
theorem  of  the  binomial.  Finally,  about  the  same  time 
Leibnitz  has  enriched  differential  calculus  by  a  nota- 
tion which,  by  indicating  the  passage  from  the  finite  to 
the  infinitely  small,  adds  to  the  advantage  of  express- 
ing the  general  results  of  calculus  that  of  giving  the 
first  approximate  values  of  the  differences  and  of  the 
sums  of  the  quantities;  this  notation  is  adapted  of  itself 
to  the  calculus  of  partial  differentials. 

We  are  often,  led  to  expressions  which  contain  so 
many  terms  and  factors  that  the  numerical  substitutions 
are  impracticable.  This  takes  place  in  questions  of 
probability  when  we  consider  a  great  number  of  events. 
Meanwhile  it  is  necessary  to  have  the  numerical  value 
of  the  formulae  in  order  to  know  with  what  probability 
the  results  are  indicated,  which  the  events  develop  by 
multiplication.  It  is  necessary  especially  to  have  the 


THE  CALCULUS  OF  PROBABILITIES.  47 

law  according  to  which  this  probability  continually 
approaches  certainty,  which  it  will  finally  attain  if  the 
number  of  events  were  infinite.  In  order  to  obtain  this 
law  I  considered  that  the  definite  integrals  of  differen- 
tials multiplied  by  the  factors  raised  to  great  powers 
would  give  by  integration  the  formulae  composed  of 
a  great  number  of  terms  and  factors.  This  remark 
brought  me  to  the  idea  of  transforming  into  similar 
integrals  the  complicated  expressions  of  analysis  and 
the  integrals  of  the  equation  of  differences.  I  fulfilled 
this  condition  by  a  method  which  gives  at  the  same 
time  the  function  comprised  under  the  integral  sign 
and  the  limits  of  the  integration.  It  offers  this  remark- 
able thing,  that  the  function  is  the  same  discriminant 
function  of  the  expressions  and  the  proposed  equations ; 
this  attaches  this  method  to  the  theory  of  discriminant 
functions  of  which  it  is  thus  the  complement.  Further, 
it  would  only  be  a  question  of  reducing  the  definite 
integral  to  a  converging  series.  This  I  have  obtained 
by  a  process  which  makes  the  series  converge  with  as 
much  more  rapidity  as  the  formula  which  it  represents 
is  "nore  complicated,  so  that  it  is  more  exact  as  it 
becomes  more  necessary.  Frequently  the  series  has 
for  a  factor  the  square  root  of  the  ratio  of  the  circum- 
ference to  the  diameter;  sometimes  it  depends  upon 
other  transcendents  whose  number  is  infinite. 

An  important  remark  which  pertains  to  great  gen- 
erality of  analysis,  and  which  permits  us  to  extend  this 
method  to  formulae  and  to  equations  of  difference  which 
the  theory  of  probability  presents  most  frequently,  is 
that  the  series  to  which  one  comes  by  supposing  the 
limits  of  the  definite  integrals  to  be  real  and  positive 


48        A  PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

take  place  equally  in  the  case  where  the  equation  which 
determines  these  limits  has  only  negative  or  imaginary 
roots.  These  passages  from  the  positive  to  the  nega- 
tive and  from  the  real  to  the  imaginary,  of  which  I  first 
have  made  use,  have  led  me  further  to  the  values  of 
many  singular  definite  integrals,  which  I  have  accord- 
ingly demonstrated  directly.  We  may  then  consider 
these  passages  as  a  means  of  discovery  parallel  to 
induction  and  analogy  long  employed  by  geometricians, 
at  first  with  an  extreme  reserve,  afterwards  with  entire 
confidence,  since  a  great  number  of  examples  has 
justified  its  use.  In  the  mean  time  it  is  always  necessary 
to  confirm  by  direct  demonstrations  the  results  obtained 
by  these  divers  means. 

I  have  named  the  ensemble  of  the  preceding  methods 
the  Calculus  of  Discriminant  Functions;  this  calculus 
serves  as  a  basis  for  the  work  which  I  have  published 
under  the  title  of  the  Analytical  Theory  of  Probabilities. 
It  is  connected  with  the  simple  idea  of  indicating  the 
repeated  multiplications  of  a  quantity  by  itself  or  its 
entire  and  positive  powers  by  writing  toward  the  top  of 
the  letter  which  expresses  it  the  numbers  which  mark 
the  degrees  of  these  powers. 

This  notation,  employed  by  Descartes  in  his  Geometry 
and  generally  adopted  since  the  publication  of  this 
important  work,  is  a  little  thing,  especially  when  com- 
pared with  the  theory  of  curves  and  variable  functions 
by  which  this  great  geometrician  has  established  the 
foundations  of  modern  calculus.  But  the  language  of 
analysis,  most  perfect  of  all,  being  in  itself  a  powerful 
instrument  of  discoveries,  its  notations,  especially  when 
they  are  necessary  and  happily  conceived,  are  so  many 


THE  CALCULUS  OF  PROBABILITIES.  49 

germs  of  new  calculi.      This  is  rendered  appreciable  by 
this  example. 

Wallis,  who  in  his  work  entitled  Arithmetica  Infini- 
torum,  one  of  those  which  have  most  contributed  to  the 
progress  of  analysis,  has  interested  himself  especially 
in  following  the  thread  of  induction  and  analogy,  con- 
sidered that  if  one  divides  the  exponent  of  a  letter  by 
two,  three,  etc.,  the  quotient  will  be  accordingly  the 
Cartesian  notation,  and  when  division  is  possible  the 
exponent  of  the  square,  cube,  etc.,  root  of  the  quantity 
which  represents  the  letter  raised  to  the  dividend 
exponent.  Extending  by  analogy  this  result  to  the 
case  where  division  is  impossible,  he  considered  a 
quantity  raised  to  a  fractional  exponent  as  the  root  of 
the  degree  indicated  by  the  denominator  of  this  frac- 
tion— namely,  of  the  quantity  raised  to  a  power  indi- 
cated by  the  numerator.  He  observed  then  that, 
according  to  the  Cartesian  notation,  the  multiplication 
of  two  powers  of  the  same  letter  amounts  to  adding 
their  exponents,  and  that  their  division  amounts  to 
subtracting  the  exponents  of  the  power  of  the  divisor 
from  that  of  the  power  of  the  dividend,  when  the  second 
of  these  exponents  is  greater  than  the  first.  Wallis 
extended  this  result  to  the  case  where  the  first 
exponent  is  equal  to  or  greater  than  the  second,  which 
makes  the  difference  zero  or  negative.  He  supposed 
then  that  a  negative  exponent  indicates  unity  divided 
by  the  quantity  raised  to  the  same  exponent  taken 
positively.  These  remarks  led  him  to  integrate 
generally  the  monomial  differentials,  whence  he  inferred 
the  definite  integrals  of  a  particular  kind  of  binomial 
differentials  whose  exponent  is  a  positive  integral 


50        A  PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 

number.  The  observation  then  of  the  law  of  the  num- 
bers which  express  these  integrals,  a  series  of  inter- 
polations and  happy  inductions  where  one  perceives 
the  germ  of  the  calculus  of  definite  integrals  which  has 
so  much  exercised  geometricians  and  which  is  one  of 
the  fundaments  of  my  new  Theory  of  Probabilities, 
gave  him  the  ratio  of  the  area  of  the  circle  to  the  square 
of  its  diameter  expressed  by  an  infinite  product,  which, 
when  one  stops  it,  confines  this  ratio  to  limits  more  and 
more  converging;  this  is  one  of  the  most  singular 
results  in  analysis.  But  it  is  remarkable  that  Wallis, 
who  had  so  well  considered  the  fractional  exponents 
of  radical  powers,  should  have  continued  to  note  these 
powers  as  had  been  done  before  him.  Newton  in  his 
Letters  to  Oldembourg,  if  I  am  not  mistaken,  was  the 
first  to  employ  the  notation  of  these  powers  by  frac- 
tional exponents.  Comparing  by  the  way  of  induction, 
of  which  Wallis  had  made  such  a  beautiful  use,  the 
exponents  of  the  powers  of  the  binomial  with  the 
coefficients  of  the  terms  of  its  development  in  the  case 
where  this  exponent  is  integral  and  positive,  he  deter- 
mined the  law  of  these  coefficients  and  extended  k  by 
analogy  to  fractional  and  negative  powers.  These 
various  results,  based  upon  the  notation  of  Descartes, 
show  his  influence  on  the  progress  of  analysis.  It  has 
still  the  advantage  of  giving  the  simplest  and  fairest 
idea  of  logarithms,  which  are  indeed  only  the  exponents 
of  a  magnitude  whose  successive  powers,  increasing  by 
infinitely  small  degrees,  can  represent  all  numbers. 

But  the  most  important  extension  that  this  notation 
has  received  is  that  of  variable  exponents,  which  con- 
stitutes exponential  calculus,  one  of  the  most  fruitful 


THE  CALCULUS  OF  PROBABILITIES.  5r 

branches  of  modern  analysis.  Leibnitz  was  the  first 
to  indicate  the  transcendents  by  variable  exponents,  and 
thereby  he  has  completed  the  system  of  elements  of 
which  a  finite  function  can  be  composed;  for  every 
finite  explicit  function  of  a  variable  may  be  reduced  in 
the  last  analysis  to  simple  magnitudes,  combined  by 
the  method  of  addition,  subtraction,  multiplication,  and 
division  and  raised  to  constant  or  variable  powers. 
The  roots  of  the  equations  formed  from  these  elements 
are  the  implicit  functions  of  tile  variable.  It  is  thus 
that  a  variable  has  for  a  logarithm  the  exponent  of  the 
power  which  is  equal  to  it  hi  the  series  of  the  powers 
of  the  number  whose  hyperbolic  logarithm  is  unity,  and 
the  logarithm  of  a  variable  of  it  is  an  implicit  function. 
Leibnitz  thought  to  give  to  his  differential  character 
the  same  exponents  as  to  magnitudes ;  but  then  in  place 
of  indicating  the  repeated  multiplications  of  the  same 
magnitude  these  exponents  indicate  the  repeated  differ- 
entiations of  the  same  function.  This  new  extension 
of  the  Cartesian  notation  led  Leibnitz  to  the  analogy  of 
positive  powers  with  the  differentials,  and  the  negative 
powers  with  the  integrals.  Lagrange  has  followed  this 
singular  analogy  in  all  its  developments;  and  by  series 
of  inductions  which  may  be  regarded  as  one  of  the 
most  beautiful  applications  which  have  ever  been  made 
of  the  method  of  induction  he  has  arrived  at  general 
formula  which  are  as  curious  as  useful  on  the  trans- 
formations of  differences  and  of  integrals  the  ones  into 
the  others  when  the  variables  have  divers  finite  incre- 
ments and  when  these  increments  are  infinitely  small. 
But  he  has  not  given  the  demonstrations  of  it  which 
appear  to  him  difficult.  The  theory  of  discriminant 


52        A  PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 

functions  extends  the  Cartesian  notations  to  some  of  its 
characters;  it  shows  with  proof  the  analogy  of  the 
powers  and  operations  indicated  by  these  characters; 
so  that  it  may  still  be  regarded  as  the  exponential 
calculus  of  characters.  All  that  concerns  the  series  and 
the  integration  of  equations  of  differences  springs  from 
it  with  an  extreme  facility. 


PART  II. 

APPLICATIONS  OF  THE  CALCULUS    OF 
PROBABILITIES. 


CHAPTER   VI. 

GAMES  OF  CHANCE. 

THE  combinations  which  games  present  were  the 
object  of  the  first  investigations  of  probabilities.  In  an 
infinite  variety  of  these  combinations  many  of  them 
lend  themselves  readily  to  calculus ;  others  require  more 
difficult  calculi;  and  the  difficulties  increasing  in  the 
measure  that  the  combinations  become  more  compli- 
cated, the  desire  to  surmount  them  and  curiosity  have 
excited  geometricians  to  perfect  more  and  more  this 
kind  of  analysis.  It  has  been  seen  already  that  the 
benefits  of  a  lottery  are  easily  determined  by  the  theory 
of  combinations.  But  it  is  more  difficult  to  know  in 
how  many  draws  one  can  bet  one  against  one,  for 
example  that  all  the  numbers  will  be  drawn,  n  being 
the  number  of  numbers,  r  that  of  the  numbers  drawn 
at  each  draw,  and  i  the  unknown  number  of  draws. 
The  expression  of  the  probability  of  drawing  all  the 

53 


54        A  PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

numbers  depends  upon  the  «th  finite  difference  of  the  i 
power  of  a  product  of  r  consecutive  numbers.  When 
the  number  n  is  considerable  the  search  for  the  value 
of  /  which  renders  this  probability  equal  to  J  becomes 
impossible  at  least  unless  this  difference  is  converted 
into  a  very  converging  series.  This  is  easily  done  by 
the  method  here  below  indicated  by  the  approxima- 
tions of  functions  of  very  large  numbers.  It  is  found 
thus  since  the  lottery  is  composed  of  ten  thousand 
numbers,  one  of  which  is  drawn  at  each  draw,  that 
there  is  a  disadvantage  in  betting  one  against  one  that 
all  the  numbers  will  be  drawn  in  95767  draws  and  an 
advantage  in  making  the  same  bet  for  95768  draws. 
In  the  lottery  of  France  this  bet  is  disadvantageous  for 
85  draws  and  advantageous  for  86  draws. 

Let  us  consider  again  two  players,  A  and  B,  playing 
together  at  heads  and  tails  in  such  a  manner  that  at 
each  throw  if  heads  turns  up  A  gives  one  counter  to  B, 
who  gives  him  one  if  tails  turns  up;  the  number  of 
counters  of  B  is  limited,  while  that  of  A  is  unlimited, 
and  the  game  is  to  end  only  when  B  shall  have  no  more 
counters.  We  ask  in  how  many  throws  one  should  bet 
one  to  one  that  the  game  will  end.  The  expression 
of  the  probability  that  the  game  will  end  in  an  i  number 
of  throws  is  given  by  a  series  which  comprises  a  great 
number  of  terms  and  factors  if  the  number  of  counters 
of  B  is  considerable;  the  search  for  the  value  of  the 
unknown  i  which  renders  this  series  \  would  then  be 
impossible  if  we  did  not  reduce  the  same  to  a  very 
convergent  series.  In  applying  to  it  the  method  of 
which  we  have  just  spoken,  we  find  a  very  simple 
expression  for  the  unknown  from  which  it  results  that  if, 


GAMES  OF  CHANCE.  55 

for  example,  B  has  a  hundred  counters,  it  is  a  bet  of  a 
little  less  than  one  against  one  that  the  game  will  end 
in  23780  throws,  and  a  bet  of  a  little  more  than  one 
against  one  that  it  will  end  in  23781  throws. 

These  two  examples  added  to  those  we  have  already 
given  are  sufficient  to  shows  how  the  problems  of 
games  have  contributed  to  the  perfection  of  analysis. 


CHAPTER  VII. 

CONCERNING  THE  UNKNOWN  INEQUALITIES 
WHICH  MAY  EXIST  AMONG  CHANCES  WHICH 
ARE  SUPPOSED  EQUAL 

INEQUALITIES  of  this  kind  have  upon  the  results  of 
the  calculation  of  probabilities  a  sensible  influence 
which  deserves  particular  attention.  Let  us  take  the 
game  of  heads  and  tails,  and  let  us  suppose  that  it  is 
equally  easy  to  throw  the  one  or  the  other  side  of  the 
coin.  Then  the  probability  of  throwing  heads  at  the 
first  throw  is  £  and  that  of  throwing  it  twice  in  succes- 
sion is  J.  But  if  there  exist  in  the  coin  an  inequality 
which  causes  one  of  the  faces  to  appear  rather  than  the 
other  without  knowing  which  side  is  favored  by  this 
inequality,  the  probability  of  throwing  heads  at  the  first 
throw  will  always  be  £;  because  of  our  ignorance  of 
which  face  is  favored  by  the  inequality  the  probability 
of  the  simple  event  is  increased  if  this  inequality  is 
favorable  to  it,  just  so  much  is  it  diminished  if  the 
inequality  is  contrary  to  it.  But  in  this  same  ignorance 
the  probability  of  throwing  heads  twice  in  succession  is 
increased.  Indeed  this  probability  is  that  of  throwing 
heads  at  the  first  throw  multiplied  by  the  probability 

56 


UNKNOWN  INEQUALITIES  AMONG   CHANCES.         57 

that  having  thrown  it  at  the  first  throw  it  will  be  thrown 
at  the  second ;  but  its  happening  at  the  first  throw  is  a 
reason  for  belief  that  the  inequality  of  the  coin  favors  it; 
the  unknown  inequality  increases,  then,  the  probability 
of  throwing  heads  at  the  second  throw ;  it  consequently 
increases  the  product  of  these  two  probabilities.  In 
order  to  submit  this  matter  to  calculus  let  us  suppose 
that  this  inequality  increases  by  a  twentieth  the  prob- 
ability of  the  simple  event  which  it  favors.  If  this 
event  is  heads,  its  probability  will  be  £  plus  -fo,  or  \±, 
and  the  probability  of  throwing  it  twice  in  succession 
will  be  the  square  of  -j^-,  or  |f  £.  If  the  favored  event  is 
tails,  the  probability  of  heads,  will  be  |  minus  ^¥>  or  *V 
and  the  probability  of  throwing  it  twice  in  succession 
will  be  TVo-  Since  we  have  at  first  no  reason  for 
believing  that  the  inequality  favors  one  of  these  events 
rather  than  the  other,  it  is  clear  that  in  order  to  have 
the  probability  of  the  compound  event  heads  heads  it 
is  necessary  to  add  the  two  preceding  probabilities  and 
take  the  half  of  their  sum,  which  gives  ^£|  for  this 
probability,  which  exceeds  £  by  ^J-g-  or  by  the  square  of 
the  favor  -fa  that  the  inequality  adds  to  the  possibilities 
of  the  event  which  it  favors.  The  probability  of  throw- 
ing tails  tails  is  similarly  £f^,  but  the  probability  of 
throwing  heads  tails  or  tails  heads  is  each  jVT;  for 
the  sum  of  these  four  probabilities  ought  to  equal  cer- 
tainty or  unity.  We  find  thus  generally  that  the 
constant  and  unknown  causes  which  favor  simple  events 
which  are  judged  equally  possible  always  increase 
the  probability  of  the  repetition  of  the  same  simple 
event. 

In  an  even  number  of  throws  heads  and  tails  ought 


58        A  PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

both  to  happen  either  an  even  number  of  times  or  odd 
number  of  times.  The  probability  of  each  of  these 
cases  is  £  if  the  possibilities  of  the  two  faces  are  equal ; 
but  if  there  is  between  them  an  unknown  inequality,  this 
inequality  is  always  favorable  to  the  first  case. 

Two  players  whose  skill  is  supposed  to  be  equal  play 
on  the  conditions  that  at  each  throw  that  one  who  loses 
gives  a  counter  to  his  adversary,  and  that  the  gam* 
continues  until  one  of  the  players  has  no  more  counters. 
The  calculation  of  the  probabilities  shows  us  that  for 
the  equality  of  the  play  the  throws  of  the  players  ought 
to  be  an  inverse  ratio  to  their  counters.  But  if  there  is 
between  the  players  a  small  unknown  inequality,  it 
favors  that  one  of  the  players  who  has  the  smallest 
number  of  counters.  His  probability  of  winning  the 
game  increases  if  the  players  agree  to  double  or  triple 
their  counters;  and  it  will  be  £  or  the  same  as  the 
probability  of  the  other  player  in  the  case  where  the 
number  of  their  counters  should  become  infinite,  pre- 
serving always  the  same  ratio. 

One  may  correct  the  influence  of  these  unknown 
inequalities  by  submitting  them  themselves  to  the 
chances  of  hazard.  Thus  at  the  play  of  heads  and 
tails,  if  one  has  a  second  coin  which  is  thrown  each 
time  with  the  first  and  one  agrees  to  name  constantly 
heads  the  face  turned  up  by  the  second  coin,  the  prob- 
ability of  throwing  heads  twice  in  succession  with  the 
first  coin  will  approach  much  nearer  \  than  in  the  case 
of  a  single  coin.  In  this  last  case  the  difference  is  the 
square  of  the  small  increment  of  possibility  that  the 
unknown  inequality  gives  to  the  face  of  the  first  coin 
which  it  favors ;  in  the  other  case  this  difference  is  the 


UNKNOWN  INEQUALITIES  AMONG   CHANCES.         59 

quadruple  product  of  this  square  by  the  corresponding 
square  relative  to  the  second  coin. 

Let  there  be  thrown  into  an  urn  a  hundred  numbers 
from  i  to  100  in  the  order  of  numeration,  and  after 
having  shaken  the  urn  in  order  to  mix  the  numbers  one 
is  drawn;  it  is  clear  that  if  the  mixing  has  been  well 
done  the  probabilities  of  the  drawing  of  the  numbers 
will  be  the  same.  But  if  we  fear  that  there  is  among 
them  small  differences  dependent  upon  the  order 
according  to  which  the  numbers  have  been  thrown  into 
the  urn,  we  shall  diminish  considerably  these  differences 
by  throwing  into  a  second  urn  the  numbers  according 
to  the  order  of  their  drawing  from  the  first  urn,  and  by 
shaking  then  this  second  urn  in  order  to  mix  the 
numbers.  A  third  urn,  a  fourth  urn,  etc.,  would 
diminish  more  and  more  these  differences  already 
inappreciable  in  the  second  urn. 


CHAPTER   VIII. 

CONCERNING  THE  LAWS  OF  PROBABILITY 
WHICH  RESULT  FROM  THE  INDEFINITE  MUL- 
TIPLICATION OF  EVENTS. 

AMID  the  variable  and  unknown  causes  which  we 
comprehend  under  the  name  of  chance,  and  which 
render  uncertain  and  irregular  the  march  of  events,  we 
see  appearing,  in  the  measure  that  they  multiply,  a 
striking  regularity  which  seems  to  hold  to  a  design  and 
which  has  been  considered  as  a  proof  of  Providence. 
But  in  reflecting  upon  this  we  spon  recognize  that  this 
regularity  is  only  the  development  of  the  respective 
possibilities  of  simple  events  which  ought  to  present 
themselves  more  often  when  they  are  more  probable. 
Let  us  imagine,  for  example,  an  urn  which  contains 
white  balls  and  black  balls;  and  let  us  suppose  that 
each  time  a  ball  is  drawn  it  is  put  back  into  the  urn 
before  proceeding  to  a  new  draw.  The  ratio  of  the 
number  of  the  white  balls  drawn  to  the  number  of  black 
balls  drawn  will  be  most  often  very  irregular  in  the  first 
drawings;  but  the  variable  causes  of  this  irregularity 
produce  effects  alternately  favorable  and  unfavorable  to 
the  regular  march  of  events  which  destroy  each  other 

60 


INDEFINITE  MULTIPLICATION  OF  EVENTS.  61 

mutually  in  the  totality  of  a  great  number  of  draws, 
allowing  us  to  perceive  more  and  more  the  ratio  cf 
white  balls  to  the  black  balls  contained  in  the  urn,  or 
the  respective  possibilities  of  drawing  a  white  ball  or 
black  ball  at  each  draw.  From  this  results  the  follow- 
ing theorem. 

The  probability  that  the  ratio  of  the  number  of  white 
balls  drawn  to  the  total  number  of  balls  drawn  does 
not  deviate  beyond  a  given  interval  from  the  ratio  of 
the  number  of  white  balls  to  the  total  number  of  balls 
contained  in  the  urn,  approaches  indefinitely  to  certainty 
by  the  indefinite  multiplication  of  events,  however  small 
this  interval. 

This  theorem  indicated  by  common  sense  was  diffi- 
cult to  demonstrate  by  analysis.  Accordingly  the 
illustrious  geometrician  Jacques  Bernouilli,  who  first 
has  occupied  himself  with  it,  attaches  great  importance 
to  the  demonstrations  he  has  given.  The  calculus  of 
discriminant  functions  applied  to  this  matter  not  only 
demonstrates  with  facility  this  theorem,  but  still  more  it 
gives  the  probability  that  the  ratio  of  the  events 
observed  deviates  only  in  certain  limits  from  the  true 
ratio  of  their  respective  possibilities. 

One  may  draw  from  the  preceding  theorem  this 
consequence  which  ought  to  be  regarded  as  a  general 
law,  namely,  that  the  ratios  of  the  acts  of  nature  are 
very  nearly  constant  when  these  acts  are  considered  in 
great  number.  Thus  in  spite  of  the  variety  of  years 
the  sum  of  the  productions  during  a  considerable  num- 
ber of  years  is  sensibly  the  same ;  so  that  man  by  useful 
foresight  is  able  to  provide  against  the  irregularity  of 
the  seasons  by  spreading  out  equally  over  all  the 


62        A  PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 

seasons  the  goods  which  nature  distributes  in  an 
unequal  manner.  I  do  not  except  from  the  above  law 
results  due  to  moral  causes.  The  ratio  of  annual 
births  to  the  population,  and  that  of  marriages  to  births, 
show  only  small  variations;  at  Paris  the  number  of 
annual  births  is  almost  the  same,  and  I  have  heard  it 
said  at  the  post-office  in  ordinary  seasons  the  number 
of  letters  thrown  aside  on  account  of  defective  addresses 
changes  little  each  year ;  this  has  likewise  been  observed 
at  London. 

It  follows  again  from  this  theorem  that  in  a  series  of 
events  indefinitely  prolonged  the  action  of  regular  and 
constant  causes  ought  to  prevail  in  the  long  run  over 
that  of  irregular  causes.  It  is  this  which  renders  the 
gains  of  the  lotteries  just  as  certain  as  the  products  of 
agriculture ;  the  chances  which  they  reserve  assure  them 
a  benefit  in  the  totality  of  a  great  number  of  throws. 
Thus  favorable  and  numerous  chances  being  constantly 
attached  to  the  observation  of  the  eternal  principles  of 
reason,  of  justice,  and  of  humanity  which  establish  and 
maintain  societies,  there  is  a  great  advantage  in  con- 
forming to  these  principles  and  of  grave  inconvenience 
in  departing  from  them.  If  one  consult  histories  and 
his  own  experience,  one  will  see  all  the  facts  come  to 
the  aid  of  this  result  of  calculus.  Consider  the  happy 
effects  of  institutions  founded  upon  reason  and  the 
natural  rights  of  man  among  the  peoples  who  have 
known  how  to  establish  and  preserve  them.  Consider 
again  the  advantages  which  good  faith  has  procured  for 
the  governments  who  have  made  it  the  basis  of  their 
conduct  and  how  they  have  been  indemnified  for  the 
sacrifices  which  a  scrupulous  exactitude  in  keeping 


INDEFINITE  MULTIPLICATION  OF  EVENTS.  63 

their  engagements  has  cost  them.  What  immense 
credit  at  home !  What  preponderance  abroad !  On 
the.  contrary,  look  into  what  an  abyss  of  misfortunes 
nations  have  often  been  precipitated  by  the  ambition 
and  the  perfidy  of  their  chiefs.  Every  time  that  a 
great  power  intoxicated  by  the  love  of  conquest  aspires 
to  universal  domination  the  sentiment  of  independence 
produces  among  the  menaced  nations  a  coalition  of 
which  it  becomes  almost  always  the  victim.  Similarly 
in  the  midst  of  the  variable  causes  which  extend  or 
restrain  the  divers  states,  the  natural  limits  acting  as 
constant  causes  ought  to  end  by  prevailing.  It  is 
important  then  to  the  stability  as  well  as  to  the  happi- 
ness of  empires  not  to  extend  them  beyond  those  limits 
into  which  they  are  led  again  without  cessation  by  the 
action  of  the  causes;  just  as  the  waters  of  the  seas 
raised  by  violent  tempests  fall  again  into  their  basins 
by  the  force  of  gravity.  It  is  again  a  result  of  the 
calculus  of  probabilities  confirmed  by  numerous  and 
melancholy  experiences.  History  treated  from  the 
point  of  view  of  the  influence  of  constant  causes  would 
unite  to  the  interest  of  curiosity  1hat  of  offering  to  man 
most  useful  lessons.  Sometimes  we  attribute  the 
inevitable  results  of  these  causes  to  the  accidental  cir- 
cumstances which  have  produced  their  action.  It  is, 
for  example,  against  the  nature  of  things  that  one 
people  should  ever  be  governed  by  another  when  a 
vast  sea  or  a  great  distance  separates  them.  It  may 
be  affirmed  that  in  the  long  run  this  constant  cause, 
joining  itself  without  ceasing  to  the  variable  causes 
which  act  in  the  same  way  and  which  the  course  of 
time  develops,  will  end  by  finding  them  sufficiently 


64        A  PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 

strong  to  give  to  a  subjugated  people  its  natural  inde- 
pendence or  to  unite  it  to  a  powerful  state  which  may 
be  contiguous. 

In  a  great  number  of  cases,  and  these  are  the  most 
important  of  the  analysis  of  hazards,  the  possibilities  of 
simple  events  are  unknown  and  we  are  forced  to  search 
in  past  events  for  the  indices  which  can  guide  us  in 
our  conjectures  about  the  causes  upon  which  they 
depend.  In  applying  the  analysis  of  discriminant 
functions  to  the  principle  elucidated  above  on  the  prob- 
ability of  the  causes  drawn  from  the  events  observed, 
we  are  led  to  the  following  theorem. 

When  a  simple  event  or  one  composed  of  several 
simple  events,  as,  for  instance,  in  a  game,  has  been 
repeated  a  great  number  of  times  the  possibilities  of  the 
simple  events  which  render  most  probable  that  which 
has  been  observed  are  those  that  observation  indicates 
with  the  greatest  probability;  in  the  measure  that  the 
observed  event  is  repeated  this  probability  increases 
and  would  end  by  amounting  to  certainty  if  the  num- 
bers of  repetitions  should  become  infinite. 

There  are  two  kinds  of  approximations:  the  one  is 
relative  to  the  limits  taken  on  all  sides  of  the  possibili- 
ties which  give  to  the  past  the  greatest  probability;  the 
other  approximation  is  related  to  the  probability  that 
these  possibilities  fall  within  these  limits.  The  repeti- 
tion of  the  compound  event  increases  more  and  more 
this  probability,  the  limits  remaining  the  same;  it 
reduces  more  and  more  the  interval  of  these  limits,  the 
probability  remaining  the  same ;  in  infinity  this  interval 
becomes  zero  and  the  probability  changes  to  certainty. 

If  we  apply  this  theorem  to  the  ratio  of  the  births  of 


INDEFINITE  MULTIPLICATION   OF  EVENTS.  65 

boys  to  that  of  girls  observed  in  the  different  countries 
of  Europe,  we  find  that  this  ratio,  which  is  everywhere 
about  equal  to  that  of  22  to  21,  indicates  with  an 
extreme  probability  a  greater  facility  in  the  birth  of 
boys.  Considering  further  that  it  is  the  same  at  Naptes 
and  at  St.  Petersburg,  we  shall  see  that  in  this  regard 
the  influence  of  climate  is  without  effect.  We  might 
then  suspect,  contrary  to  the  common  belief,  that  this 
predominance  of  masculine  births  exists  even  in  the 
Orient.  I  have  consequently  invited  the  French 
scholars  sent  to  Egypt  to  occupy  themselves  with  this 
interesting  question ;  but  the  difficulty  in  obtaining 
exact  information  about  the  births  has  not  permitted 
them  to  solve  it.  Happily,  M.  de  Humboldt  has  not 
neglected  this  matter  among  the  innumerable  new 
things  which  he  has  observed  and  collected  in  America 
with  so  much  sagacity,  constancy,  and  courage.  He 
has  found  in  the  tropics  the  same  ratio  of  the  births  as 
we  observe  in  Paris ;  this  ought  to  make  us  regard  the 
greater  number  of  masculine  births  as  a  general  law  of 
the  human  race.  The  laws  which  the  different  kinds 
of  animals  follow  in  this  regard  seem  to  me  worthy  of 
the  attention  of  naturalists. 

The  fact  that  the  ratio  of  births  of  boys  to  that  of 
girls  differs  very  little  from  unity  even  in  the  great 
number  of  the  births  observed  in  a  place  would  offer  in 
this  regard  a  result  contrary  to  the  general  law,  without 
which  we  should  be  right  in  concluding  that  this  law 
did  not  exist.  In  order  to  arrive  at  this  result  it  is 
necessary  to  employ  great  numbers  and  to  be  sure  that 
it  is  indicated  by  great  probability.  Buffon  cites,  for 
example,  in  his  Political  AritJimctic  several  communi- 


66        A  PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 

ties  of  Bourgogne  where  the  births  of  girls  have  sur- 
passed those  of  boys.  Among  these  communities  that 
of  Carcelle-le-Grignon  presents  in  20x39  births  during 
five  years  1026  girls  and  983  boys.  Although  these 
numbers  are  considerable,  they  indicate,  however,  only 
a  greater  possibility  in  the  births  of  girls  with  a  prob- 
ability of  -fa,  and  this  probability,  smaller  than  that  cf 
not  throwing  heads  four  times  in  succession  in  the  game 
of  heads  and  tails,  is  not  sufficient  to  investigate  the 
cause  for  this  anomaly,  which,  according  to  all  prob- 
ability, would  disappear  if  one  should  follow  during  a 
century  ihe  births  in  this  community. 

The  registers  of  births,  which  are  kept  with  care  in 
order  to  assure  the  condition  of  the  citizens,  may  serve 
in  determining  the  population  of  a  great  empire  without 
recurring  to  the  enumeration  of  its  inhabitants — a 
laborious  operation  and  one  difficult  to  make  with 
exactitude.  But  for  this  it  is  necessary  to  know  the 
ratio  of  the  population  to  the  annual  births.  The  most 
precise  means  of  obtaining  it  consists,  first,  in  choosing 
in  the  empire  districts  distributed  in  an  almost  equal 
manner  over  its  whole  surface,  so  as  to  render  the 
general  result  independent  of  local  circumstances; 
second,  in  enumerating  with  care  for  a  given  epoch  the 
inhabitants  of  several  communities  in  each  of  these  dis- 
tricts; third,  by  determining  from  the  statement  of  the 
births  during  several  years  which  precede  and  follow 
this  epoch  the  mean  number  corresponding  to  the 
annual  births.  This  number,  divided  by  that  of  the 
inhabitants,  will  give  the  ratio  of  the  annual  births  to 
the  population  in  a  manner  more  and  more  accurate 
as  the  enumeration  becomes  more  considerable.  The 


INDEFINITE  MULTIPLICATION  OF  EVENTS.  67 

government,  convinced  of  the  utility  of  a  similar 
enumeration,  has  decided  at  my  request  to  order 
its  execution.  In  thirty  districts  spread  out  equally 
over  the  whole  of  France,  communities  have  been 
chosen  which  would  be  able  to  furnish  the  most  exact 
information.  Their  enumerations  have  given  2037615 
individuals  as  the  total  number  of  their  inhabitants  on 
the  23d  of  September,  1802.  The  statement  of  the 
births  in  these  communities  during  the  years  1800, 
1 80 1,  and  1802  have  given: 

Births.  Marriages.  Deaths. 

1 103 1 2  boys         46037          103659  men 
105287  girls  99443  women 

The  ratio  of  the  population  to  annual  births  is 
then  28T3I75oWb7r>  ^  is  greater  than  had  been  estimated 
up  to  this  time.  Multiplying  the  number  of  annual 
births  in  France  by  this  ratio,  we  shall  have  the  pop- 
ulation of  this  kingdom.  But  what  is  the  probability 
that  the  population  thus  determined  will  not  deviate 
from  the  true  population  beyond  a  given  limit  ? 
Resolving  this  problem  and  applying  to  its  solution  the 
preceding  data,  I  have  found  that,  the  number  of  annual 
births  in  France  being  supposed  to  be  1000000,  which 
brings  the  population  to  28352845  inhabitants,  it  is  a 
bet  of  almost  300000  against  I  that  the  error  of  this 
result  is  not  half  a  million. 

The  ratio  of  the  births  of  boys  to  that  of  girls  which 
the  preceding  statement  offers  is  that  of  22  to  21  ;  and 
the  marriages  are  to  the  births  as  3  is  to  4. 

At  Paris  the  baptisms  of  children  of  both  sexes  vary 
a  little  from  the  ratio  of  22  to  21.  Since  1745,  the 


68        A  PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 

epoch  in  which  one  has  commenced  to  distinguish  the 
sexes  upon  the  birth-registers,  up  to  the  end  of  1/84, 
there  have  been  baptized  in  this  capital  393386  boys 
and  377555  girls.  The  ratio  of  the  two  numbers  is 
almost  that  of  2  5  to  24 ;  it  appears  then  at  Paris  that  a 
particular  cause  approximates  an  equality  of  baptisms 
of  the  two  sexes.  If  we  apply  to  this  matter  the 
calculus  of  probabilities,  we  find  that  it  is  a  bet  of  238 
to  i  in  favor  of  the  existence  of  this  cause,  which  is 
sufficient  to  authorize  the  investigation.  Upon  reflec- 
tion it  has  appeared  to  me  that  the  difference  observed 
holds  to  this,  that  the  parents  in  the  country  and  the 
provinces,  finding  some  advantage  in  keeping  the  boys 
at  home,  have  sent  to  the  Hospital  for  Foundlings  in 
Paris  fewer  of  them  relative  to  the  number  of  girls 
according  to  the  ratio  of  births  of  the  two  sexes.  This 
is  proved  by  the  statement  of  the  registers  of  this 
hospital.  From  the  beginning  of  1745  to  the  end  of 
1809  there  were  entered  163499  boys  and  159405 
girls.  The  first  of  these  numbers  exceeds  only  by  -$$ 
the  second,  which  it  ought  to  have  surpassed  at  least 
by  ?V-  This  confirms  the  existence  of  the  assigned 
cause,  namely,  that  the  ratio  of  births  of  boys  to  those 
of  girls  is  at  Paris  that  of  22  to  21,  no  attention  having 
been  paid  to  foundlings. 

The  preceding  results  suppose  that  we  may  compare 
the  births  to  the  drawings  of  balls  from  an  urn  which 
contains  an  infinite  number  of  white  balls  and  black 
balls  so  mixed  that  at  each  draw  the  chances  of  drawing 
ought  to  be  the  same  for  each  ball;  but  it  is  possible 
that  the  variations  of  the  same  seasons  in  different 
years  may  have  some  influence  upon  the  annual  ratio 


INDEFINITE  MULTIPLICATION  Of-'  EVENTS.  69 

of  the  births  of  boys  to  those  of  girls.  The  Bureau  of 
Longitudes  of  France  publishes  each  year  in  its  annual 
the  tables  of  the  annual  movement  of  the  population  of 
the  kingdom.  The  tables  already  published  commence 
in  1817;  in  that  year  and  in  the  five  following  years 
there  were  born  2962361  boys  and  2781997  girls, 
which  gives  about  T£  for  the  ratio  of  the  births  of  boys 
to  that  of  girls.  The  ratios  of  each  year  vary  little 
from  this  mean  result;  the  smallest  ratio  is  that  of 
1822,  where  it  was  only  ff ;  the  greatest  is  of  the  year 
1817,  when  it  was  ff.  These  ratios  vary  appreciably 
from  the  ratio  of  |f  found  above.  Applying  to  this 
deviation  the  analysis  of  probabilities  in  the  hypothesis 
of  the  comparison  of  births  to  the  drawings  of  balls 
from  an  urn,  we  find  that  it  would  be  scarcely  probable. 
It  appears,  then,  to  indicate  that  this  hypothesis, 
although  closely  approximated,  is  not  rigorously  exact. 
In  the  number  of  births  which  we  have  just  stated  there 
are  of  natural  children  200494  boys  and  190698  girls. 
The  ratio  of  masculine  and  feminine  births  was  then  in 
this  regard  ff ,  smaller  than  the  mean  ratio  of  ff .  This 
result  is  in  the  same  sense  as  that  of  the  births  of 
foundlings;  and  it  seems  to  prove  that  in  the  class  of 
natural  children  the  births  of  the  two  sexes  approach 
more  nearly  equality  than  in  the  class  of  legitimate 
children.  The  difference  of  the  climates  from  the  north 
to  the  south  of  France  does  not  appear  to  influence 
appreciably  the  ratio  of  the  births  of  boys  and  girls. 
The  thirty  most  southern  districts  have  given  T|  for  this 
ratio,  the  same  as  that  of  entire  France. 

The  constancy  of  the  superiority  of  the  births  of  boys 
over  girls  at  Paris  and  at  London  since  they  have  been 


?o        A   PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

observed  has  appeared  to  some  scholars  to  be  a  proof 
of  Providence,  without  which  they  have  thought  that 
the  irregular  causes  which  disturb  without  ceasing  the 
course  of  events  ought  several  times  to  have  rendered 
the  annual  births  of  girls  superior  to  those  of  boys. 

But  this  proof  is  a  new  example  of  the  abuse  which 
has  been  so  often  made  of  final  causes  which  always 
disappear  on  a  searching  examination  of  the  questions 
when  we  have  the  necessary  data  to  solve  them.  The 
constancy  in  question  is  a  result  of  regular  causes  which 
give  the  superiority  to  the  births  of  boys  and  which 
extend  it  to  the  anomalies  due  to  hazard  when  the 
number  of  annual  births  is  considerable.  The  investi- 
gation of  the  probability  that  this  constancy  will  main- 
tain itself  for  a  long  time  belongs  to  that  branch  of  the 
analysis  of  hazards  which  passes  from  past  events  to 
the  probability  of  future  events ;  and  taking  as  a  basis 
the  births  observed  from  1745  to  1784,  it  is  a  bet  of 
almost  4  against  I  that  at  Paris  the  annual  births  of 
boys  will  constantly  surpass  for  a  century  the  births 
of  girls ;  there  is  then  no  reason  to  be  astonished  that 
this  has  taken  place  for  a  half-century. 

Let  us  take  another  example  of  the  development  of 
constant  ratios  which  events  present  in  the  measure 
that  they  are  multiplied.  Let  us  imagine  a  series  of 
urns  arranged  circularly,  and  each  containing  a  very 
great  number  of  white  balls  and  black  balls ;  the  ratio 
of  white  balls  to  the  black  in  the  urns  being  originally 
very  different  and  such,  for  example,  that  one  of  these 
urns  contains  only  white  balls,  while  another  contains 
only  black  balls.  If  one  draws  a  ball  from  the  first  urn 
in  order  to  put  it  into  the  second,  and,  after  having 


INDEFINITE  MULTIPLICATION  OF  EVENTS.  7* 

shaken  the  second  urn  in  order  to  mix  well  the  new 
ball  with  the  others,  one  draws  a  ball  to  put  it  into  the 
third  urn,  and  so  on  to  the  last  urn,  from  which  is  drawn 
a  ball  to  put  into  the  first,  and  if  this  series  is  recom- 
menced continually,  the  analysis  of  probability  shows 
us  that  the  ratios  of  the  white  balls  to  the  black  in  these 
urns  will  end  by  being  the  same  and  equal  to  the  ratio 
of  the  sum  of  all  the  white  balls  to  the  sum  of  all  the 
black  balls  contained  in  the  urns.  Thus  by  this  regular 
mode  of  change  the  primitive  irregularity  of  these  ratios 
disappears  eventually  in  order  to  make  room  for  the 
most  simple  order.  Now  if  among  these  urns  one 
intercalate  new  ones  in  which  the  ratio  of  the  sum  of 
the  white  balls  to  the  sum  of  the  black  balls  which  they 
contain  differs  from  the  preceding,  continuing  indefi- 
nitely in  the  totality  of  the  urns  the  drawings  which  we 
have  just  indicated,  the  simple  order  established  in  the 
old  urns  will  be  at  first  disturbed,  and  the  ratios  of  the 
white  balls  to  the  black  balls  will  become  irregular; 
but  little  by  little  this  irregularity  will  disappear  in 
order  to  make  room  for  a  new  order,  which  will  finally 
be  that  of  the  equality  of  the  ratios  of  the  white  balls 
to  the  black  balls  contained  in  the  urns.  We  may 
apply  these  results  to  all  the  combinations  of  nature  in 
which  the  constant  forces  by  which  their  elements  are 
animated  establish  regular  modes  of  action,  suited  to 
bring  about  in  the  very  heart  of  chaos  systems  governed 
by  admirable  laws. 

The  phenomena  which  seem  the  most  dependent 
upon  hazard  present,  then,  when  multiplied  a  tendency 
to  approach  without  ceasing  fixed  ratios,  in  such  a 
manner  that  if  we  conceive  on  all  sides  of  each  of  these 


72        A  PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 

ratios  an  interval  as  small  as  desired,  the  probability 
that  the  mean  result  of  the  observations  falls  within  this 
interval  will  end  by  differing  from  certainty  only  by  a 
quantity  greater  than  an  assignable  magnitude.  Thus 
by  the  calculations  of  probabilities  applied  to  a  great 
number  of  observations  we  may  recognize  the  existence 
of  these  ratios.  But  before  seeking  the  causes  it  is 
necessary,  in  order  not  to  be  led  into  vain  speculations, 
to  assure  ourselves  that  they  are  indicated  by  a  prob- 
ability which  does  not  permit  us  to  regard  them  as 
anomalies  due  to  hazard.  The  theory  of  discriminant 
functions  gives  a  very  simple  expression  for  this  prob- 
ability, which  is  obtained  by  integrating  the  product  of 
the  differential  of  the  quantity  of  which  the  result 
deduced  from  a  great  number  of  observations  varies 
from  the  truth  by  a  constant  less  than  unity,  dependent 
upon  the  nature  of  the  problem,  and  raised  to  a  power 
whose  exponent  is  the  ratio  of  the  square  of  this  varia- 
tion to  the  number  of  observations.  The  integral  taken 
between  the  limits  given  and  divided  by  the  same 
integral,  applied  to  a  positive  and  negative  infinity, 
will  express  the  probability  that  the  variation  from  the 
truth  is  comprised  between  these  limits.  Such  is  the 
general  law  of  the  probability  of  results  indicated  by  a 
great  number  of  observations. 


CHAPTER    IX. 

THE  APPLICATION  OF  THE  CALCULUS  OF  PROB- 
ABILITIES  TO  NATURAL  PHILOSOPHY. 

THE  phenomena  of  nature  are  most  often  enveloped 
by  so  many  strange  circumstances,  and  so  great  a 
number  of  disturbing  causes  mix  their  influence,  that 
it  is  very  difficult  to  recognize  them.  We  may  arrive 
at  them  only  by  multiplying  the  observations  or  the 
experiences,  so  that  the  strange  effects  finally  destroy 
reciprocally  each  other,  the  mean  results  putting  in 
evidence  those  phenomena  and  their  divers  elements. 
The  more  numerous  the  number  of  observations  and 
the  less  they  vary  among  themselves  the  more  their 
results  approach  the  truth.  We  fulfil  this  last  condition 
by  the  choice  of  the  methods  of  observations,  by  the 
precision  of  the  instruments,  and  by  the  care  which  we 
take  to  observe  closely;  then  we  determine  by  the 
theory  of  probabilities  the  most  advantageous  mean 
results  or  those  which  give  the  least  value  of  the  error. 
But  that  is  not  sufficient;  it  is  further  necessary  to 
appreciate  the  probability  that  the  errors  of  these 
results  are  comprised  in  the  given  limits;  and  without 
this  we  have  only  an  imperfect  knowledge  of  the  degree 


74        A  PHILOSOPHICAL   ESSAY   ON  PROBABILITIES. 

of  exactitude  obtained.  Formulas  suitable  to  these 
matters  are  then  true  improvements  of  the  method  of 
sciences,  and  it  is  indeed  important  to  add  them  to  this 
method.  The  analysis  which  they  require  is  the  most 
delicate  and  the  most  difficult  of  the  theory  of  prob-^ 
abilities;  it  is  one  of  the  principal  objects  of  the  work 
which  I  have  published  upon  this  theory,  and  in  which 
I  have  arrived  at  formulas  of  this  kind  which  have  the 
remarkable  advantage  of  being  independent  of  the  law 
of  the  probability  of  errors  and  of  including  only  the 
quantities  given  by  the  observations  themselves  and 
their  expressions. 

Each  observation  has  for  an  analytic  expression  a 
function  of  the  elements  which  we  wish  to  determine; 
and  if  these  elements  are  nearly  known,  this  function 
becomes  a  linear  function  of  their  corrections.  In 
equating  it  to  the  observation  itself  there  is  formed  an 
equation  of  condition.  If  we  have  a  great  number  of 
similar  equations,  we  combine  them  in  such  a  manner 
as  to  obtain  as  many  final  equations  as  there  are  ele- 
ments whose  corrections  we  determine  then  by  resolv- 
ing these  equations.  But  what  is  the  most  advantageous 
manner  of  combining  equations  of  condition  in  order 
to  obtain  final  equations  ?  What  is  the  law  of  the 
probabilities  of  errors  of  which  the  elements  are  still 
susceptible  that  we  draw  from  them  ?  This  is  made 
clear  to  us  by  the  theory  of  probabilities.  The  forma- 
tion of  a  final  equation  by  means  of  the  equation  of 
condition  amounts  to  multiplying  each  one  of  these  by 
an  indeterminate  factor  and  by  uniting  the  products;  it 
is  necessary  to  choose  the  system  of  factors  which  gives 
the  smallest  opportunity  for  error.  But  it  is  apparent 


PROBABILITIES  AND  NATURAL   PHILOSOPHY.         75 

that  if  we  multiply  the  possible  errors  of  an  element  by 
their  respective  probabilities,  the  most  advantageous 
system  will  be  that  in  which  the  sum  of  these  products 
all,  taken,  positively  is  a  minimum;  for  a  positive  or  a 
negative  error  ought  to  be  considered  as  a  loss.  Form- 
ing, then,  this  sum  of  products,  the  condition  of  the 
minimum  will  determine  the  system  of  factors  which  it 
is  expedient  to  adopt,  or  the  most  advantageous  system. 
We  find  thus  that  this  system  is  that  of  the  coefficients 
of  the  elements  in  each  equation  of  condition ;  so  that 
we  form  a  first  final  equation  by  multiplying  respect- 
ively each  equation  of  condition  by  its  coefficient  of 
the  first  element  and  by  uniting  all  these  equations  thus 
multiplied.  We  form  a  second  final  equation  by  em- 
ploying in  the  same  manner  the  coefficients  of  tl.e 
second  element,  and  so  on.  In  this  manner  the  ele- 
ments and  the  laws  of  the  phenomena  obtained  in  the 
collection  of  a  great  number  of  observations  are 
developed  with  the  most  evidence. 

The  probability  of  the  errors  which  each  element 
still  leaves  to  be  feared  is  proportional  to  the  number 
whose  hyperbolic  logarithm  is  unity  raised  to  a  power 
equal  to  the  square  of  the  error  taken  as  a  minus 
quantity  and  multiplied  by  a  constant  coefficient  which 
may  be  considered  as  the  modulus  of  the  probability  of 
the  errors ;  because,  the  error  remaining  the  same,  its 
probability  decreases  with  rapidity  when  the  former 
increases;  so  that  the  element  obtained  weighs,  if  I 
may  thus  speak  toward  the  truth,  as  much  more  as  this 
modulus  is  greater.  I  would  call  for  this  reason  this 
modulus  the  weigJit  of  the  element  or  of  the  result. 
This  weight  is  the  greatest  possible  in  the  system  of 


76        A  PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

factors — the  most  advantageous;  it  is  this  which  gives 
to  this  system  superiority  over  others.  By  a  remarkable 
analogy  of  this  weight  with  those  of  bodies  compared 
at  their  common  centre  of  gravity  it  results  that  if  the 
same  element  is  given  by  divers  systems,  composed 
each  of  a  great  number  of  observations,  the  most 
advantageous,  the  mean  result  of  their  totality  is  the 
sum  of  the  products  of  each  partial  result  by  its  weight. 
Moreover,  the  total  weight  of  the  results  of  the  divers 
systems  is  the  sum  of  their  partial  weights ;  so  that  the 
probability  of  the  errors  of  the  mean  result  of  their 
totality  is  proportional  to  the  number  which  has  unity 
for  an  hyperbolic  logarithm  raised  to  a  power  equal  to 
the  square  of  the  error  taken  as  minus  and  multiplied 
by  the  sum  of  the  weights.  Each  weight  depends  in 
truth  upon  the  law  of  the  probability  of  error  of  each 
system,  and  almost  always  this  law  is  unknown;  but 
happily  I  have  been  able  to  eliminate  the  factor  which 
contains  it  by  means  of  the  sum  of  the  squares  of  the 
variations  of  the  observations  in  this  system  from  their 
mean  result.  It  would  then  be  desirable  in  order  to 
complete  our  knowledge  of  the  results  obtained  by  the 
totality  of  a  great  number  of  observations  that  we  write 
by  the  side  of  each  result  the  weight  which  corresponds 
to  it;  analysis  furnishes  for  this  object  both  general  and 
simple  methods.  When  we  have  thus  obtained  the 
exponential  which  represents  the  law  of  the  proba- 
bility of  errors,  we  shall  have  the  probability  that  the 
error  of  the  result  is  included  within  given  limits  by 
taking  within  the  limits  the  integral  of  the  product  of 
this"  exponential  by  the  differential  of  the  error  and 
multiplying  it  by  the  square  root  of  the  weight  of  the 


PROBABILITIES  AND   NATURAL   PHILOSOPHY.          77 

result  divided  by  the  circumference  whose  diameter  is 
unity.  Hence  it  follows  that  for  the  same  probability 
the  errors  of  the  results  are  reciprocal  to  the  square 
roots  of  their  weights,  which  serves  to  compare  their 
respective  precision. 

In  order  to  apply  this  method  with  success  it  is 
necessary  to  vary  the  circumstances  of  the  observations 
or  the  experiences  in  such  a  manner  as  to  avoid  the 
constant  causes  of  error.  It  is  necessary  that  the 
observations  should  be  numerous,  and  that  they  should 
be  so  much  the  more  so  as  there  are  more  elements  to 
determine ;  for  the  weight  of  the  mean  result  increases 
as  the  number  of  observations  divided  by  the  number 
of  the  elements.  It  is  still  necessary  that  the  elements 
follow  in  these  observations  a  different  course;  for  if  the 
course  of  the  two  elements  were  exactly  the  same, 
which  would  render  their  coefficients  proportional  in 
equation  of  conditions,  these  elements  would  form  only 
a  single  unknown  quantity  and  it  would  be  impossible 
to  distinguish  them  by  these  observations.  Finally  it 
is  necessary  that  the  observations  should  be  precise; 
this  condition,  the  first  of  all,  increases  greatly  the 
weight  of  the  result  the  expression  of  which  has  for 
a  divisor  the  sum  of  the  squares  of  the  deviations  of  the 
observations  from  this  result.  With  these  precautions 
we  shall  be  able  to  make  use  of  the  preceding  method 
and  measure  the  degree  of  confidence  which  the  results 
deduced  from  a  great  number  of  observations  merit. 

The  rule  which  we  have  just  given  to  conclude  equa- 
tions of  condition,  final  equations,  amount  to  rendering 
a  minimum  the  sum  of  the  squares  of  the  errors  of 
observations;  for  each  equation  of  condition  becomes 


78        A  PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

exact  by  substituting  in  it  the  observation  plus  its 
error;  and  if  we  draw  from  it  the  expression  of  this 
error,  it  is  easy  to  see  that  the  condition  of  the  minimum 
of  the  sum  of  the  squares  of  these  expressions  gives  the 
rule  in  question.  This  rule  is  the  more  precise  as  the 
observations  are  more  numerous ;  but  even  in  the  case 
where  their  number  is  small  it  appears  natural  to 
employ  the  same  rule  which  in  all  cases  offers  a  simple 
means  of  obtaining  without  groping  the  corrections 
which  we  seek  to  determine.  It  serves  further  to  com- 
pare the  precision  of  the  divers  astronomical  tables  of 
the  same  star.  These  tables  may  always  be  supposed 
as  reduced  to  the  same  form,  and  then  they  differ  only 
by  the  epochs,  the  mean  movements  and  the  coefficients 
of  the  arguments ;  for  if  one  of  them  contains  a  coeffi- 
cient which  is  not  found  in  the  others,  it  is  clear  that 
this  amounts  to  supposing  zero  in  them  as  the  coefficient 
of  this  argument.  If  now  we  rectify  these  tables  by 
the  totality  of  the  good  observations,  they  would  satisfy 
the  condition  that  the  sum  of  the  squares  of  the  errors 
should  be  a  minimum;  the  tables  which,  compared  to  a 
considerable  number  of  observations,  approach  nearest 
this  condition  merit  then  the  preference. 

It  is  principally  in  astronomy  that  the  method 
explained  above  may  be  employed  with  advantage. 
The  astronomical  tables  owe  the  truly  astonishing 
exactitude  which  they  have  attained  to  the  precision  of 
observations  and  of  theories,  and  to  the  use  of  equations 
of  conditions  which  cause  to  concur  a  great  number  of 
excellent  observations  in  the  correction  of  the  same 
element.  But  it  remains  to  determine  the  probability 
of  the  errors  that  this  correction  leaves  still  to  be 


PROBABILITIES  AND  NATURAL   PHILOSOPHY.         79 

feared ;  and  the  method  which  I  have  just  explained 
enables  us  to  recognize  the  probability  of  these  errors. 
In  order  to  give  some  interesting  applications  of  it  I 
have  profited  by  the  immense  work  which  M.  Bouvard 
has  just  finished  on  the  movements  of  Jupiter  and 
Saturn,  of  which  he  has  formed  very  precise  tables. 
He  has  discussed  with  the  greatest  care  the  oppositions 
and  quadratures  of  these  two  planets  observed  by 
Bradley  and  by  the  astronomers  who  have  followed 
him  down  to  the  last  years;  he  has  concluded  the  cor- 
rections of  the  elements  of  their  movement  and  their 
masses  compared  to  that  of  the  sun  taken  as  unity. 
His  calculations  give  him  the  mass  of  Saturn  equal  to 
the  3512th  part  of  that  of  the  sun.  Applying  to  them 
my  formulas  of  probability,  I  find  that  it  is  a  bet  of 
n,ooo  against  one  that  the  error  of  this  result  is  not 
T^  of  its  value,  or  that  which  amounts  to  almost  the 
same — that  after  a  century  of  new  observations  added  to 
the  preceding  ones,  and  examined  in  the  same  manner, 
the  new  result  will  not  differ  by  TLff  from  that  of 
M.  Bouvard.  This  wise  astronomer  finds  again  the 
mass  of  Jupiter  equal  to  the  ro/ith  part  of  the  sun; 
and  my  method  of  probability  gives  a  bet  of  1,000,000 
to  one  that  this  result  is  not  T^o-  in  error. 

This  method  may  be  employed  again  with  success  in 
geodetic  operations.  We  determine  the  length  of  the 
great  arc  on  the  surface  of  the  earth  by  triangulation, 
which  depends  upon  a  base  measured  with  exactitude. 
But  whatever  precision  may  be  brought  to  the  measure 
of  the  angles,  the  inevitable  errors  can,  by  accumulat- 
ing, cause  the  value  of  the  arc  concluded  from  a  great 
number  of  triangles  to  deviate  appreciably  from  the 


8o        A  PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

truth.  We  recognize  this  value,  then,  only  imperfectly 
unless  the  probability  that  its  error  is  comprised  within 
given  limits  can  be  assigned.  The  error  of  a  geodetic 
result  is  a  function  of  the  errors  of  the  angles  of  each 
triangle.  I  have  given  in  the  work  cited  general 
formulae  in  order  to  obtain  the  probability  of  the  values 
of  one  or  of  several  linear  functions  of  a  great  number 
of  partial  errors  of  which  we  know  the  law  of  prob- 
ability; we  may  then  by  means  of  these  formulae  deter- 
mine the  probability  that  the  error  of  a  geodetic  result 
is  contained  within  the  assigned  limits,  whatever  may  be 
the  law  of  the  probability  of  partial  errors.  It  is  more- 
over more  necessary  to  render  ourselves  independent 
of  the  law,  since  the  most  simple  laws  themselves  are 
always  infinitely  less  probable,  seeing  the  infinite 
number  of  those  which  may  exist  in  nature.  But  the 
unknown  law  of  partial  errors  introduces  into  the 
formula::  an  indeterminant  which  does  not  permit  of 
reducing  them  to  numbers  unless  we  are  able  to  elimi- 
nate it.  We  have  seen  that  in  astronomical  questions, 
where  each  observation  furnishes  an  equation  of  condi- 
tion for  obtaining  the  elements,  we  eliminate  this 
determinant  by  means  of  the  sum  of  the  squares  of  the 
remainders  when  the  most  probable  values  of  the  ele- 
ments have  been  substituted  in  each  equation.  Geodetic 
questions  not  offering  similar  equations,  it  is  necessary 
to  seek  another  means  of  elimination.  The  quantity 
by  which  the  sum  of  the  angles  of  each  observed  tri- 
angle surpasses  two  right  angles  plus  the  spherical 
excess  furnishes  this  means.  Thus  we  replace  by  the 
sum  of  the  squares  of  these  quantities  the  sum  of  the 
squares  of  the  remainders  of  the  equations  of  condition ; 


PROBABILITIES  AND  NATURAL  PHILOSOPHY.         81 

and  we  may  assign  in  numbers  the  probability  that  the 
error  of  the  final  result  of  a  series  of  geodetic  operations 
will  not  exceed  a  given  quantity.  But  what  is  the 
most  advantageous  manner  of  dividing  among  the  three 
angles  of  each  triangle  the  observed  sum  of  their 
errors  ?  The  analysis  of  probabilities  renders  it 
apparent  that  each  angle  ought  to  be  diminished  by  a 
third  of  this  sum,  provided  that  the  weight  of  a  geodetic 
result  be  the  greatest  possible,  which  renders  the  same 
error  less  probable.  There  is  then  a  great  advantage 
in  observing  the  three  angles  of  each  triangle  and  of 
correcting  them  as  we  have  just  said.  Simple  common 
sense  indicates  this  advantage;  but  the  calculation  of 
probabilities  alone  is  able  to  appreciate  it  and  to  render 
apparent  that  by  this  correction  it  becomes  the  greatest 
possible. 

In  order  to  assure  oneself  of  the  exactitude  of  the 
value  of  a  great  arc  which  rests  upon  a  base  measured 
at  one  of  its  extremities  one  measures  a  second  base 
toward  the  other  extremity;  and  one  concludes  from 
one  of  these  bases  the  length  of  the  other.  If  this 
length  varies  very  little  from  the  observation,  there  is 
all  reason  to  believe  that  the  chain  of  triangles  which 
unites  these  bases  is  very  nearly  exact  and  likewise  the 
value  of  the  large  arc  which  results  from  it.  One  cor- 
rects, then,  this  value  by  modifying  the  angles  of  the 
triangles  in  such  a  manner  that  the  base  is  calculated 
according  to  the  bases  measured.  But  this  may  be 
done  in  an  infinity  of  ways,  among  which  is  preferred 
that  of  which  the  geodetic  result  has  the  greatest 
weight,  inasmuch  as  the  same  error  becomes  less  prob- 
able. The  analysis  of  probabilities  gives  formulae  for 


82        A  PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

obtaining  directly  the  most  advantageous  correction 
which  results  from  the  measurements  of  the  several 
bases  and  the  laws  of  probability  which  the  multiplicity 
of  the  bases  makes — laws  which  become  very  rapidly 
decreasing  by  this  multiplicity. 

Generally  the  errors  of  the  results  deduced  from  a 
great  number  of  observations  are  the  linear  functions 
of  the  partial  errors  of  each  observation.  The  coeffi- 
cients of  these  functions  depend  upon  the  nature  of  the 
problem  and  upon  the  process  followed  in  order  to 
obtain  the  results.  The  most  advantageous  process  is 
evidently  that  in  which  the  same  error  in  the  results  is 
less  probable  than  according  to  any  other  process. 
The  application  of  the  calculus  of  probabilities  to 
natural  philosophy  consists,  then,  in  determining  analyti- 
cally the  probability  of  the  values  of  these  functions 
and  in  choosing  their  indeterminant  coefficients  in  such 
a  manner  that  the  law  of  this  probability  should  be 
most  rapidly  descending.  Eliminating,  then,  from  the 
formulae  by  the  data  of  the  question  the  factor  which  is 
introduced  by  the  almost  always  unknown  law  of  the 
probability  of  partial  errors,  we  may  be  able  to  evaluate 
numerically  the  probability  that  the  errors  of  the  results 
do  not  exceed  a  given  quantity.  We  shall  thus  have 
all  that  may  be  desired  touching  the  results  deduced 
from  a  great  number  of  observations. 

Very  approximate  results  may  be  obtained  by  other 
considerations.  Suppose,  for  example,  that  one  has  a 
thousand  and  one  observations  of  the  same  quantity; 
the  arithmetical  mean  of  all  these  observations  is  the 
result  given  by  the  most  advantageous  method.  But 
one  would  be  able  to  choose  the  result  according  to  the 


PROBABILITIES  AND  NATURAL   PHILOSOPHY.         83 

condition  that  the  sum  of  the  variations  from  each 
partial  value  all  taken  positively  should  be  a  minimum. 
It  appears  indeed  natural  to  regard  as  very  approximate 
the  result  which  satisfies  this  condition.  It  is  easy  to 
see  that  if  one  disposes  the  values  given  by  the  obser- 
vations according  to  the  order  of  magnitude,  the  value 
which  will  occupy  the  mean  will  fulfil  the  preceding 
condition,  and  calculus  renders  it  apparent  that  in  the 
case  of  an  infinite  number  of  observations  it  would 
coincide  with  the  truth;  but  the  result  given  by  the 
most  advantageous  method  is  still  preferable. 

We  see  by  that  which  precedes  that  the  theory  of 
probabilities  leaves  nothing  arbitrary  in  the  manner  of 
distributing  the  errors  of  the  observations;  it  gives  for 
this,  distribution  the  most  advantageous  formulae  which 
diminishes  as  much  as  possible  the  errors  to  be  feared 
in  the  results. 

The  consideration  of  probabilities  can  serve  to  dis- 
tinguish the  small  irregularities  of  the  celestial  move- 
ments enveloped  in  the  errors  of  observations,  and  to 
repass  to  the  cause  of  the  anomalies  observed  in  these 
movements. 

In  comparing  all  the  observations  it  was  Ticho-Brahe 
who  recognized  the  necessity  of  applying  to  the  moon 
an  equation  of  time  different  from  that  which  had  been 
applied  to  the  sun  and  to  the  planets.  It  was  similarly 
the  totality  of  a  great  number,  of  observations  which 
made  Mayer  recognize  that  the  coefficient  of  the 
inequality  of  the  precession  ought  to  be  diminished  a 
little  for  the  moon.  But  since  this  diminution,  although 
confirmed  and  even  augmented  by  Mason,  did  not 
appear  to  result  from  universal  gravitation,  the  majority 


84        A  PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

of  astronomers  neglect  it  in  their  calculations.  Having 
submitted  to  the  calculation  of  probabilities  a  consider- 
able number  of  lunar  observations  chosen  for  this 
purpose  and  which  M.  Bouvard  consented  to  examine 
at  my  request,  it  appeared  to  me  to  be  indicated  with 
so  strong  a  probability  that  I  believed  the  cause  of  it 
ought  to  be  investigated.  I  soon  saw  that  it  would  be 
only  the  ellipticity  of  the  terrestrial  spheroid,  neglected 
up  to  that  time  in  the  theory  of  the  lunar  movement  as 
being  able  to  produce  only  imperceptible  terms.  I 
concluded  that  these  terms  became  perceptible  by  the 
successive  integrations  of  differential  equations.  I 
determined  then  those  terms  by  a  particular  analysis, 
and  I  discovered  first  the  inequality  of  the  lunar  move- 
ment in  latitude  which  is  proportional  to  the  sine  of 
the  longitude  of  the  moon,  which  no  astronomer  before 
had  suspected.  I  recognized  then  by  means  of  this 
inequality  that  another  exists  in  the  lunar  movement  in 
longitude  which  produces  the  diminution  observed  by 
Mayer  in  the  equation  of  the  precession  applicable  1o 
the  moon.  The  quantity  of  this  diminution  and  the 
coefficient  of  the  preceding  inequality  in  latitude  are 
very  appropriate  to  fix  the  oblateness  of  the  earth. 
Having  communicated  my  researches  to  M.  Burg,  who 
was  occupied  at  that  time  in  perfecting  the  tables  of 
the  moon  by  the  comparison  of  all  the  good  observa- 
tions, I  requested  him  to  determine  with  a  particular 
care  these  two  quantities.  By  a  very  remarkable 
agreement  the  values  which  he  has  found  give  to  the 
earth  the  same  oblateness,  7JT,  which  differs  little  from 
the  mean  derived  from  the  measurements  of  the  degrees 
of  the  meridian  and  the  pendulum ;  but  those  regarded 


PROBABILITIES  AND  NATURAL   PHILOSOPHY.         85 

from  the  point  of  view  of  the  influence  of  the  errors  of 
the  observations  and  of  the  perturbing  causes  in  these 
-measurements,  did  not  appear  to  me  exactly  determined 
by  these  lunar  inequalities. 

It  was  again  by  the  consideration  of  probabilities  that 
I  recognized  the  cause  of  the  secular  equation  of  the 
moon.  The  modern  observations  of  this  star  compared 
to  the  ancient  eclipses  had  indicated  to  astronomers  an 
acceleration  in  the  lunar  movement ;  but  the  geometri- 
cians, and  particularly  Lagrange,  having  vainly  sought 
in  the  perturbations  which  this  movement  experienced 
the  terms  upon  which  this  acceleration  depends,  reject 
it.  An  attentive  examination  of  the  ancient  and 
modern  observations  and  of  the  intermediary  eclipses 
observed  by  the  Arabians  convinced  me  that  it  was 
indicated  with  a  great  probability.  I  took  up  again 
then  from  this  point  of  view  the  lunar  theory,  and  I 
recognized  that  the  secular  equation  of  the  moon  is  due 
to  the  action  of  the  sun  upon  this  satellite,  combined 
with  the  secular  variation  of  the  eccentricity  of  the  ter- 
restrial orb ;  this  brought  me  to  the  discovery  of  the 
secular  equations  of  the  movements  of  the  nodes  and 
of  the  perigees  of  the  lunar  orbit,  which  equations  had 
not  been  even  suspected  by  astronomers.  The  very 
remarkable  agreement  of  this  theory  with  all  the 
ancient  and  modern  observations  has  brought  it  to  a 
very  high  degree  of  evidence. 

The  calculus  of  probabilities  has  led  me  similarly  to 
the  cause  of  the  great  irregularities  of  Jupiter  and 
Saturn.  Comparing  modern  observations  with  ancient, 
Halley  found  an  acceleration  in  the  movement  of 
Jupiter  and  a  retardation  in  that  of  Saturn.  In  order 


86        A  PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

to  conciliate  the  observations  he  reduced  the  move- 
ments to  two  secular  equations  of  contrary  signs  and 
increasing  as  the  squares  of  the  times  passed  since 
1700.  Euler  and  Lagrange  submitted  to  analysis  the 
alterations  which  the  mutual  attraction  of  these  two 
planets  ought  to  produce  in  these  movements.  They 
found  in  doing  this  the  secular  equations;  but  their 
results  were  so  different  that  one  of  the  two  at  least 
ought  to  be  erroneous.  I  determined  then  to  take  up 
again  this  important  problem  of  celestial  mechanics,  and 
I  recognized  the  invariability  of  the  mean  planetary 
movements,  which  nullified  the  secular  equations  intro- 
duced by  Halley  in  the  tables  of  Jupiter  and  Saturn. 
Thus  there  remain,  in  order  to  explain  the  great 
irregularity  of  these  planets,  only  the  attractions  of  the 
comets  to  which  many  astronomers  had  effective 
recourse,  or  the  existence  of  an  irregularity  over  a  long 
period  produced  in  the  movements  of  the  two  planets 
by  their  reciprocal  action  and  affected  by  contrary 
signs  for  each  of  them.  A  theorem  which  I  found  in 
regard  to  the  inequalities  of  this  kind  rendered  this 
inequality  very  probable.  According  to  this  theorem, 
if  the  movement  of  Jupiter  is  accelerated,  that  of  Saturn 
is  retarded,  which  has  already  conformed  to  what 
Halley  had  noticed;  moreover,  the  acceleration  of 
Jupiter  resulting  from  the  same  theorem  is  to  the 
retardation  of  Saturn  very  nearly  in  the  ratio  of  the 
secular  equations  proposed  by  Halley.  Considering  the 
mean  movements  of  Jupiter  and  Saturn  I  was  enabled 
easily  to  recognize  that  two  times  that  of  Jupiter 
differed  only  by  a  very  small  quantity  from  five  times 
that  of  Saturn.  The  period  of  an  irregularity  which 


PROBABILITIES  AND  NATURAL  PHILOSOPHY.         87 

would  have  for  an  argument  this  difference  would  be 
about  nine  centuries.  Indeed  its  coefficient  would  be 
of  the  order  of  the  cubes  of  the  eccentricities  of  the 
orbits;  but  I  knew  that  by  virtue  of  successive  integra- 
tions it  acquired  for  divisor  the  square  of  the  very  small 
multiplier  of  the  time  in  the  argument  of  this  inequality 
which  is  able  to  give  it  a  great  value ;  the  existence  of 
this  inequality  appeared  to  me  then  very  probable. 
The  following  observation  increased  then  its  probability. 
Supposing  its  argument  zero  toward  the  epoch  of  the 
observations  of  Ticho-Brahe,  I  saw  that  Halley  ought 
to  have  found  by  the  comparison  of  modern  with  ancient 
observations  the  alterations  which  he  had  indicated ; 
while  the  comparison  of  the  modern  observations  among 
themselves  ought  to  offer  contrary  alterations  similar 
to  those  which  Lambert  had  concluded  from  this  com- 
parison. I  did  not  then  hesitate  at  all  to  undertake 
this  long  and  tedious  calculation  necessary  to  assure 
myself  of  this  inequality.  It  was  entirely  confirmed  by 
the  result  of  this  calculation,  which  moreover  made  me 
recognize  a  great  number  of  other  inequalities  of  which 
the  totality  has  inclined  the  tables  of  Jupiter  and  Saturn 
to  the  precision  of  the  same  observations. 

It  was  again  by  means  of  the  calculus  of  probabilities 
that  I  recognized  the  remarkable  law  of  the  mean 
movements  of  the  three  first  satellites  of  Jupiter,  accord- 
ing to  which  the  mean  longitude  of  the  first  minus 
three  times  that  of  the  second  plus  two  times  that  of 
the  third  is  rigorously  equal  to  the  half-circumference. 
The  approximation  with  which  the  mean  movements  of 
these  stars  satisfy  this  law  since  their  discovery  indicates 
its  existence  with  an  extreme  probability.  I  sought 


88        A  PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 

then  the  cause  of  it  in  their  mutual  action.  The 
searching  examination  of  this  action  convinced  me  that 
it  was  sufficient  if  in  the  beginning  the  ratios  of  their 
mean  movements  had  approached  this  law  within 
certain  limits,  because  their  mutual  action  had  estab- 
lished and  maintained  it  rigorously.  Thus  these  three 
bodies  will  balance  one  another  eternally  in  space 
according  to  the  preceding  law  unless  strange  causes, 
such  as  comets,  should  change  suddenly  their  move- 
ments about  Jupiter. 

Accordingly  it  is  seen  how  necessary  it  is  to  be  at- 
tentive to  the  indications  of  nature  when  they  are  the 
result  of  a  great  number  of  observations,  although  in 
other  respects  they  may  be  inexplicable  by  known 
means.  The  extreme  difficulty  of  problems  relative  to 
the  system  of  the  world  has  forced  geometricians  to  recur 
to  the  approximation  which  always  leaves  room  for  the 
fear  that  the  quantities  neglected  may  have  an  appreci- 
able influence.  When  they  have  been  warned  of  this 
influence  by  the  observations,  they  have  recurred  to 
their  analysis ;  in  rectifying  it  they  have  always  found 
the  cause  of  the  anomalies  observed ;  they  have  deter- 
mined the  laws  and  often  they  have  anticipated  the 
observations  in  discovering  the  inequalities  which  it  had 
not  yet  indicated.  Thus  one  may  say  that  nature 
itself  has  concurred  in  the  analytical  perfection  of  the 
theories  based  upon  the  principle  of  universal  gravity; 
and  this  is  to  my  mind  one  of  the  strongest  proofs  of 
the  truth  of  this  admirable  principle. 

In  the  cases  which  I  have  just  considered  the 
analytical  solution  of  the  question  has  changed  the 
probability  of  the  causes  into  certainty.  But  most  often 


PROBABILITIES  AND  NATURAL   PHILOSOPHY.         89 

this  solution  is  impossible  and  it  remains  only  to 
augment  more  and  more  this  probability.  In  the  midst 
of  numerous  and  incalculable  modifications  which  the 
action  of  the  causes  receives  then  from  strange  circum- 
stances these  causes  conserve  always  with  the  effects 
observed  the  proper  ratios  to  make  them  recognizable 
and  to  verify  their  existence.  Determining  these  ratios 
and  comparing  them  with  a  great  number  of  observa- 
tions if  one  finds  that  they  constantly  satisfy  it,  the 
probability  of  the  causes  may  increase  to  the  point  of 
equalling  that  of  facts  in  regard  to  which  there  is  no 
doubt.  The  investigation  of  these  ratios  of  causes  to 
their  effects  is  not  less  useful  in  natural  philosophy 
than  the  direct  solution  of  problems  whether  it  be  to 
verify  the  reality  of  these  causes  or  to  determine  the 
laws  from  their  effects ;  since  it  may  be  employed  in  a 
great  number  of  questions  whose  direct  solution  is  not 
possible,  it  replaces  it  in  the  most  advantageous 
manner.  I  shall  discuss  here  the  application  which  I 
have  made  of  it  to  one  of  the  most  interesting  phenom- 
ena of  nature,  the  flow  and  the  ebb  of  the  sea. 

Pline  has  given  of  this  phenomenon  a  description 
remarkable  for  its  exactitude,  and  in  it  one  sees  that 
the  ancients  had  observed  that  the  tides  of  each  month 
are  greatest  toward  the  syzygies  and  smallest  toward 
the  quadratures ;  that  they  are  higher  in  the  perigees 
than  in  the  apogees  of  the  moon,  and  higher  in  the 
equinoxes  than  in  the  solstices.  They  concluded  from 
this  that  this  phenomenon  is  due  to  the  action  of  the 
sun  and  moon  upon  the  sea.  In  the  preface  of  his 
work  De  Stella  Martis  Kepler  admits  a  tendency  of  the 
waters  of  the  sea  toward  the  moon ;  but,  ignorant  of  the 


90        A  PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 

law  of  this  tendency,  he  was  able  to  give  on  this  subject 
only  a  probable  idea.  Newton  converted  into  certainty 
the  probability  of  this  idea  by  attaching  it  to  his  great 
principle  of  universal  gravity.  He  gave  the  exact 
expression  of  the  attractive  forces  which  produced  the 
flood  and  the  ebb  of  the  sea;  and  in  order  to  determine 
the  effects  he  supposed  that  the  sea  takes  at  each 
instant  the  position  of  equilibrium  which  is  agreeable 
to  these  forces.  He  explained  in  this  manner  the 
principal  phenomena  of  the  tides ;  but  it  followed  from 
this  theory  that  in  our  ports  the  two  tides  of  the  same 
day  would  be  very  unequal  if  the  sun  and  the  moon 
should  have  a  great  declination.  At  Brest,  for  exam- 
ple, the  evening  tide  would  be  in  the  syzygies  of  the 
solstices  about  eight  times  greater  than  the  morning 
tide,  which  is  certainly  contrary  to  the  observations 
which  prove  that  these  two  tides  are  very  nearly  equal. 
This  result  from  the  Newtonian  theory  might  hold  to 
the  supposition  that  the  sea  is  agreeable  at  each  instant 
to  a  position  of  equilibrium,  a  supposition  which  is  not 
at  all  admissible.  But  the  investigation  of  the  true 
figure  of  the  sea  presents  great  difficulties.  Aided  by 
the  discoveries  which  the  geometricians  had  just  made 
in  the  theory  of  the  movement  of  fluids  and  in  the 
calculus  of  partial  differences,  I  undertook  this  investi- 
gation, and  I  gave  the  differential  equations  of  the 
movement  of  the  sea  by  supposing  that  it  covers  the 
entire  earth.  In  drawing  thus  near  to  nature  I  had  the 
satisfaction  of  seeing  that  my  results  approached  the 
observations,  especially  in  regard  to  the  little  difference 
which  exists  in  our  ports  between  the  two  tides  of  the 
solstitial  syzygies  of  the  same  day.  I  found  that  they 


PROBABILITIES  AND  NATURAL   PHILOSOPHY.         91 

would  be  equal  if  the  sea  had  everywhere  the  same 
depth ;  I  found  further  that  in  giving  to  this  depth 
convenient  values  one  was  able  to  augment  the  height 
of  the  tides  in  a  port  conformably  to  the  observations. 
But  these  investigations,  in  spite  of  their  generality,  did 
not  satisfy  at  all  the  great  differences  which  even 
adjacent  ports  present  in  this  regard  and  which  prove 
the  influence  of  local  circumstances.  The  impossibility 
of  knowing  these  circumstances  and  the  irregularity  of 
the  basin  of  the  seas  and  that  of  integrating  the  equa- 
tions of  partial  differences  which  are  relative  has  com- 
pelled me  to  make  up  the  deficiency  by  the  method  I 
have  indicated  above.  I  then  endeavored  to  determine 
the  greatest  ratios  possible  among  the  forces  which 
affect  all  the  molecules  of  the  sea,  and  their  effects 
observable  in  our  ports.  For  this  I  made  use  of  the 
following  principle,  which  may  be  applied  to  many 
other  phenomena. 

' '  The  state  of  the  system  of  a  body  in  which  the 
primitive  conditions  of  the  movement  have  disappeared 
by  the  resistances  which  this  movement  meets  is 
periodic  as  the  forces  which  animate  it. ' ' 

Combining  this  principle  with  that  of  the  coexistence 
of  very  small  oscillations,  I  have  found  an  expression 
of  the  height  of  the  tides  whose  arbitraries  contain  the 
effect  of  local  cricumstances  of  each  port  and  are 
reduced  to  the  smallest  number  possible ;  it  is  only 
necessary  to  compare  it  to  a  great  number  of  observa- 
tions. 

Upon  the  invitation  of  the  Academy  of  Sciences, 
observations  were  made  at  the  beginning  of  the  last 
century  at  Brest  upon  the  tides,  which  were  continued 


92        A  PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 

during  six  consecutive  years.  The  situation  of  this 
port  is  very  favorable  to  this  sort  of  observations;  it 
communicates  with  the  sea  by  a  canal  which  empties 
into  a  vast  roadstead  at  the  far  end  of  which  the  port 
has  been  constructed.  The  irregularities  of  the  sea 
extend  thus  only  to  a  small  degree  into  the  port,  just 
as  the  oscillations  which  the  irregular  movement  of  a 
vessel  produces  in  a  barometer  are  diminished  by  a 
throttling  made  in  the  tube  of  this  instrument.  More- 
over, the  tides  being  considerable  at  Brest,  the  acciden- 
tal variations  caused  by  the  winds  are  only  feeble; 
likewise  we  notice  in  the  observations  of  these  tides, 
however  little  we  multiply  them,  a  great  regularity 
which  induced  me  to  propose  to  the  government  to 
order  in  this  port  a  new  series  of  observations  of  the 
tides,  continued  during  a  period  of  the  movement  of  the 
nodes  of  the  lunar  orbit.  This  has  been  done.  The 
observations  began  June  1 ,  1 806 ;  and  since  this  time 
they  have  been  made  every  day  without  interruption. 
I  am  indebted  to  the  indefatigable  zeal  of  M.  Bouvard, 
for  all  that  interests  astronomy,  the  immense  calcula- 
tions which  the  comparison  of  my  analysis  with  the 
observations  has  demanded.  There  have  been  used 
about  six  thousand  observations,  made  during  the  year 
1 807  and  the  fifteen  years  following.  It  results  from 
this  comparison  that  my  formulae  represent  with  a 
remarkable  precision  all  the  varieties  of  the  tides  rela- 
tive to  the  digression  of  the  moon,  from  the  sun,  to  the 
declination  of  these  stars,  to  their  distances  from  the 
earth,  and  to  the  laws  of  variation  at  the  maximum  and 
minimum  of  each  of  these  elements.  There  results 
from  this  accord  a  probability  that  the  flow  and  the  ebb 


PROBABILITIES  AND  NATURAL  PHILOSOPHY.         93 

of  the  sea  is  due  to  the  attraction  of  the  sun  and  moon, 
so  approaching  certainty  that  it  ought  to  leave  room 
for  no  reasonable  doubt.  It  changes  into  certainty 
when  we  consider  that  this  attraction  is  derived  from 
the  law  of  universal  gravity  demonstrated  by  all  the 
celestial  phenomena. 

The  action  of  the  moon  upon  the  sea  is  more  than 
double  that  of  the  sun.  Newton  and  his  successors  in 
the  development  of  this  action  have  paid  attention 
only  to  the  terms  divided  by  the  cube  of  the  distance 
from  the  moon  to  the  earth,  judging  that  the  effects 
due  to  the  following  terms  ought  to  be  inappreciable. 
But  the  calculation  of  probabilities  makes  it  clear  to  us 
that  the  smallest  effects  of  regular  causes  may  manifest 
themselves  in  the  results  of  a  great  number  of  observa- 
tions arranged  in  the  order  most  suitable  to  indicate 
them.  This  calculation  again  determines  their  prob- 
ability and  up  to  what  point  it  is  necessary  to  multiply 
the  observations  to  make  it  very  great.  Applying  it 
to  the  numerous  observations  discussed  by  M.  Bouvard 
I  recognized  that  at  Brest  the  action  of  the  moon  upon 
the  sea  is  greater  in  the  full  moons  than  in  the  new 
moons,  and  greater  when  the  moon  is  austral  than 
when  it  is  boreal — phenomena  which  can  result  only 
from  the  terms  of  the  lunar  action  divided  by  the 
fourth  power  of  the  distance  from  the  moon  to  the 
earth. 

To  arrive  at  the  ocean  the  action  of  the  sun  and  the 
moon  traverses  the  atmosphere,  which  ought  conse- 
quently to  feel  its  influence  and  to  be  subjected  to 
movements  similar  to  those  of  the  sea. 

These  movements  produce  in  the  barometer  periodic 


94        A  PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

oscillations.  Analysis  has  made  it  clear  to  me  that 
they  are  inappreciable  in  our  climates.  But  as  local 
circumstances  increase  considerably  the  tides  in  our 
ports,  I  have  inquired  again  if  similar  circumstances 
have  made  appreciable  these  oscillations  of  the 
barometer.  For  this  I  have  made  use  of  the  meteoro- 
logical observations  which  have  been  made  every  day 
for  many  years  at  the  royal  observatory.  The  heights 
of  the  barometer  and  of  the  thermometer  are  observed 
there  at  nine  o'clock  in  the  morning,  at  noon,  at  three 
o'clock  in  the  afternoon,  and  at  eleven  o'clock  in  the 
evening.  M.  Bouvard  has  indeed  wished  to  take  up 
the  consideration  of  observations  of  the  eight  years 
elapsed  from  October  I,  1815,  to  October  I,  1823,  on 
the  registers.  In  disposing  the  observations  in  the 
manner  most  suitable  to  indicate  the  lunar  atmospheric 
flood  at  Paris,  I  find  only  one  eighteenth  of  a  milli- 
meter for  the  extent  of  the  corresponding  oscillation  of 
the  barometer.  It  is  this  especially  which  has  made 
us  feel  the  necessity  of  a  method  for  determining  the 
probability  of  a  result,  and  without  this  method  one  is 
forced  to  present  as  the  laws  of  nature  the  results  of 
irregular  causes  which  has  often  happened  in  mete- 
orology. This  method  applied  to  the  preceding  result 
shows  the  uncertainty  of  it  in  spite  of  the  great  number 
of  observations  employed,  which  it  would  be  necessary 
to  increase  tenfold  in  order  to  obtain  a  result  suffi- 
ciently probable. 

The  principle  which  serves  as  a  basis  for  my  theory 
of  the  tides  may  be  extended  to  all  the  effects  of  hazard 
to  which  variable  causes  are  joined  according  to  regular 
laws.  The  action  of  these  causes  produces  in  the  mean 


PROBABILITIES  AND   NATURAL   PHILOSOPHY.          95 

results  of  a  great  number  of  effects  varieties  which 
follow  the  same  laws  and  which  one  may  recognize  by 
the  analysis  of  probabilities.  In  the  measure  which 
these  effects  are  multiplied  those  varieties  are  mani- 
fested with  an  ever-increasing  probability,  which  would 
approach  certainty  if  the  number  of  the  effects  of  the 
results  should  become  infinite.  This  theorem  is 
analogous  to  that  which  I  have  already  developed  upon 
the  action  of  constant  causes.  Every  time,  then,  that 
a  cause  whose  progress  is  regular  can  have  influence 
upon  a  kind  of  events,  we  may  seek  to  discover  its 
influence  by  multiplying  the  observations  and  arrang- 
ing them  in  the  most  suitable  order  to  indicate  it. 
When  this  influence  appears  to  manifest  itself  the 
analysis  of  probabilities  determines  the  probability  of 
its  existence  and  that  of  its  intensity ;  thus  the  variation 
of  the  temperature  from  day  to  night  modifying  the  pres- 
sure of  the  atmosphere  and  consequently  the  height  of 
the  barometer,  it  is  natural  to  think  that  the  multiplied 
observations  of  these  heights  ought  to  show  the  influ- 
ence of  the  solar  heat.  Indeed  there  has  long  been 
recognized  at  the  equator,  where  this  influence  appears 
to  be  greatest,  a  small  diurnal  variation  in  the  height 
of  the  barometer  of  which  the  maximum  occurs  about 
nine  o'clock  in  the  morning  and  the  minimum  about 
three  o'clock  in  the  afternoon.  A  second  maxivntin 
occurs  about  eleven  o'clock  in  the  evening  and  a 
second  minimum  about  four  o'clock  in  the  morning. 
The  oscillations  of  the  night  are  less  than  those  of  the 
day,  the  extent  of  which  is  about  two  millimeters. 
The  inconstancy  of  our  climate  has  not  taken  this 
variation  from  our  observers,  although  it  may  be  less 


96        A  PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 

appreciable  than  in  the  tropics.  M.  Ramond  has 
recognized  and  determined  it  at  Clermont,  the  chief 
place  of  the  district  of  Puy- de-Dome,  by  a  series  of 
precise  observations  made  during  several  years ;  he  has 
even  found  that  it  is  smaller  in  the  months  of  winter 
than  in  other  months.  The  numerous  observations 
which  I  have  discussed  in  order  to  estimate  the  influ- 
ence of  attractions  of  the  sun  and  the  moon  upon  the 
barometric  heights  at  Paris  have  served  me  in  deter- 
mining their  diurnal  variation.  Comparing  the  heights 
at  nine  o'clock  in  the  morning  with  those  of  the  same 
days  at  three  o'clock  in  the  afternoon,  this  variation  is 
manifested  with  so  much  evidence  that  its  mean  value 
each  month  has  been  constantly  positive  for  each  of 
the  seventy-two  months  from  January  I,  1817,  to 
January  I,  1823;  its  mean  value  in  these  seventy-two 
months  has  been  almost  .8  of  a  millimeter,  a  little  less 
than  at  Clermont  and  much  less  than  at  the  equator. 
I  have  recognized  that  the  mean  result  of  the  diurnal 
variations  of  the  barometer  from  9  o'clock  A.M.  to 
3  P.M.  has  been  only  .5428  millimeter  in  the  three 
months  of  November,  December,  January,  and  that  it 
has  risen  to  1.0563  millimeters  in  the  three  following 
months,  which  coincides  with  the  observations  of 
M.  Ramond.  The  other  months  offer  nothing  similar. 
In  order  to  apply  to  these  phenomena  the  calculation 
of  these  probabilities,  I  commenced  by  determining  the 
law  of  the  probability  of  the  anomalies  of  the  diurnal 
variation  due  to  hazard.  Applying  it  then  to  the  obser- 
vations of  this  phenomenon,  I  found  that  it  was  a  bet  01 
more  than  300,00x3  against  one  that  a  regular  cause 
produced  it.  I  do  not  seek  to  determine  this  cause;  I 


PROBABILITIES  AND  NATURAL   PHILOSOPHY.         97 

content  myself  with  stating  its  existence.  The  period 
of  the  diurnal  variation  regulated  by  the  solar  day  indi- 
cates evidently  that  this  variation  is  due  to  the  action 
of  the  sun.  The  extreme  smallness  of  the  attractive 
action  of  the  sun  upon  the  atmosphere  is  proved  by  the 
smallness  of  the  effects  due  to  the  united  attractions  of 
the  sun  and  the  moon.  It  is  then  by  the  action  of  its 
heat  that  the  sun  produces  the  diurnal  variation  of  the 
barometer ;  but  it  is  impossible  to  subject  to  calculus 
the  effects  of  its  action  on  the  height  of  the  barometer 
and  upon  the  winds.  The  diurnal  variation  of  the 
magnetic  needle  is  certainly  a  result  of  the  action  of 
the  sun.  But  does  this  star  act  here  as  in  the  diurnal 
variation  of  the  barometer  by  its  heat  or  by  its  influence 
upon  electricity  and  upon  magnetism,  or  finally  by  the 
union  of  these  influences  ?  A  long  series  of  observa- 
tions made  in  different  countries  will  enable  us  to 
apprehend  this. 

One  of  the  most  remarkable  phenomena  of  the 
system  of  the  world  is  that  of  all  the  movemens  of 
rotation  and  of  revolution  of  the  planets  and  the 
satellites  in  the  sense  of  the  rotation  of  the  sun  and 
about  in  the  same  plane  of  its  equator.  A  phenomenon 
so  remarkable  is  not  the  effect  of  hazard :  it  indicates 
a  general  cause  which  has  determined  all  its  move- 
ments. In  order  to  obtain  the  probability  with  which 
this  cause  is  indicated  we  shall  observe  that  the 
planetary  system,  such  as  we  know  it  to-day,  is  com- 
posed of  eleven  planets  and  of  eighteen  satellites  at 
least,  if  we  attribute  with  Herschel  six  satellites  to  the 
planet  Uranus.  The  movements  of  the  rotation  of  the 
sun,  of  six  planets,  of  the  moon,  of  the  satellites  of 


98        A  PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

Jupiter,  of  the  ring  of  Saturn,  and  of  one  of  its  satellites 
have  been  recognized.  These  movements  form  with 
those  of  revolution  a  totality  of  forty-three  movements 
directed  in  the  same  sense;  but  one  finds  by  the  analy- 
sis of  probabilities  that  it  is  a  bet  of  more  than 
4000000000000  against  one  that  this  disposition  is 
not  the  result  of  hazard ;  this  forms  a  probability  indeed 
superior  to  that  of  historical  events  in  regard  to  which 
no  doubt  exists.  We  ought  then  to  believe  at  least 
with  equal  confidence  that  a  primitive  cause  has 
directed  the  planetary  movements,  especially  if  we 
consider  that  the  inclination  of  the  greatest  number  of 
these  movements  at  the  solar  equator  is  very  small. 

Another  equally  remarkable  phenomenon  of  the  solar 
system  is  the  small  degree  of  the  eccentricity  of  the 
orbs  of  the  planets  and  the  satellites,  while  those  of  the 
comets  are  very  elongated,  the  orbs  of  the  system  not 
offering  any  intermediate  shades  between  a  great  and 
a  small  eccentricity.  We  are  again  forced  to  recog- 
nize here  the  effect  of  a  regular  cause;  chance  has 
certainly  not  given  an  almost  circular  form  to  the 
orbits  of  all  the  planets  and  their  satellites ;  it  is  then 
that  the  cause  which  has  determined  the  movements  of 
these  bodies  has  rendered  them  almost  circular.  It  is 
necessary,  again,  that  the  great  eccentricities  of  the 
orbits  of  the  comets  should  result  from  the  existence 
of  this  cause  without  its  having  influenced  the  direction 
of  their  movements ;  for  it  is  found  that  there  are  almost 
as  many  retrograde  comets  as  direct  comets,  and  that 
the  mean  inclination  of  all  their  orbits  to  the  ecliptic 
approaches  very  nearly  half  a  right  angle,  as  it  ought 
to  be  if  the  bodies  had  been  thrown  at  hazard. 


PROBABILITIES  AND  NATURAL   PHILOSOPHY.         99 

Whatever  may  be  the  nature  of  the  cause  in  question, 
since  it  has  produced  or  directed  the  movement  of  the 
planets,  it  is  necessary  that  it  should  have  embraced  all 
the  bodies  and  considered  all  the  distances  which  sepa- 
rate them,  it  can  have  been  only  a  fluid  of  an  immense 
extension.  Therefore  in  order  to  have  given  them  in 
the  same  sense  an  almost  circular  movement  about  the 
sun  it  is  necessary  that  this  fluid  should  have  surrounded 
this  star  as  an  atmosphere.  The  consideration  of  the 
planetary  movements  leads  us  then  to  think  that  by 
virtue  of  an  excessive  heat  the  atmosphere  of  the  sun 
was  originally  extended  beyond  the  orbits  of  all  the 
planets,  and  that  it  has  contracted  gradually  to  its 
present  limits. 

In  the  primitive  state  where  we  imagine  the  sun  it 
resembled  the  nebulae  that  the  telescope  shows  us 
composed  of  a  nucleus  more  or  less  brilliant  surrounded 
by  a  nebula  which,  condensing  at  the  surface,  ought 
to  transform  it  some  day  into  a  star.  If  one  conceives 
by  analogy  all  the  stars  formed  in  this  manner,  one 
can  imagine  their  anterior  state  of  nebulosity  itself  pre- 
ceded by  other  stars  in  which  the  nebulous  matter  was 
more  and  more  diffuse,  the  nucleus  being  less  and  less 
luminous  and  dense.  Going  back,  then,  as  far  as 
possible,  one  would  arrive  at  a  nebulosity  so  diffuse 
that  one  would  be  able  scarcely  to  suspect  its  exist- 
ence. 

Such  is  indeed  the  first  state  of  the  nebulae  which 
Herschel  observed  with  particular  care  by  means  of  his 
powerful  telescopes,  and  in  which  he  has  followed  the 
progress  of  condensation,  not  in  a  single  one,  these 
stages  not  becoming  appreciable  t6  us  except  after 


loo     A  PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 

centuries,  but  in  their  totality,  just  about  as  one  can  in 
a  vast  forest  follow  the  increase  of  the  trees  by  the 
individuals  of  the  divers  ages  which  the  forest  contains. 
He  has  observed  from  the  beginning  nebulous  matter 
spread  out  in  divers  masses  in  the  different  parts  of  the 
heavens,  of  which  it  occupies  a  great  extent.  He  has 
seen  in  some  of  these  masses  this  matter  slightly  con- 
densed about  one  or  several  faintly  luminous  nebulae. 
In  the  other  nebulae  these  nuclei  shine,  moreover,  in 
proportion  to  the  nebulosity  which  surrounds  them. 
The  atmospheres  of  each  nucleus  becoming  separated 
by  an  ulterior  condensation,  there  result  the  multifold 
nebulas  formed  of  brilliant  nuclei  very  adjacent  and 
surrounded  each  by  an  atmosphere;  sometimes  the 
nebulous  matter,  by  condensing  in  a  uniform  manner, 
has  produced  the  nebulae  which  are  called  planetary. 
Finally  a  greater  degree  of  condensation  transforms  all 
these  nebulae  into  stars.  The  nebulae  classed  accord- 
ing to  this  philosophic  view  indicate  with  an  extreme 
probability  their  future  transformation  into  stars  and 
the  anterior  state  of  nebulosity  of  existing  stars.  The 
following  considerations  come  to  the  aid  of  proofs 
drawn  from  these  analogies. 

For  a  long  time  the  particular  disposition  of  certain 
stars  visible  to  the  naked  eye  has  struck  the  attention 
of  philosophical  observers.  Mitchel  has  already 
remarked  how  improbable  it  is  that  the  stars  of  the 
Pleiades,  for  example,  should  have  been  confined  in 
the  narrow  space  which  contain  them  by  the  chances 
of  hazard  alone,  and  he  has  concluded  from  this  that 
this  group  of  stars  and  the  similar  groups  that  the 
heaven  presents  tis  are  the  results  of  a  primitive  cause 


PROBABILITIES  AND  NATURAL   PHILOSOPHY.       1OI 

or  of  a  general  law  of  nature.  These  groups  are  a 
necessary  result  of  the  condensation  of  the  nebulae  at 
several  nuclei ;  it  is  apparent  that  the  nebulous  matter 
being  attracted  continuously  by  the  divers  nuclei,  they 
ought  to  form  in  time  a  group  of  stars  equal  to  that  of 
the  Pleiades.  The  condensation  of  the  nebulae  at  two 
nuclei  forms  similarly  very  adjacent  stars,  revolving  the 
one  about  the  other,  equal  to  those  whose  respective 
movements  Herschel  has  already  considered.  Such 
are,  further,  the  6ist  of  the  Swan  and  its  following  one 
in  which  Bessel  has  just  recognized  particular  move- 
ments so  considerable  and  so  little  different  that  the 
proximity  of  these  stars  to  one  another  and  their 
movement  about  the  common  centre  of  gravity  ought 
to  leave  no  doubt.  Thus  one  descends  by  degrees 
from  the  condensation  of  nebulous  matter  to  the  con- 
sideration of  the  sun  surrounded  formerly  by  a  vast 
atmosphere,  a  consideration  to  which  one  repasses,  as 
has  been  seen,  by  the  examination  of  the  phenomena 
of  the  solar  system.  A  case  so  remarkable  gives  to 
the  existence  of  this  anterior  state  of  the  sun  a  prob- 
ability strongly  approaching  certainty. 

But  how  has  the  solar  atmosphere  determined  the 
movements  of  rotation  and  revolution  of  the  planets 
and  the  satellites  ?  If  these  bodies  had  penetrated 
deeply  the  atmosphere  its  resistance  would  have  caused 
them  to  fall  upon  the  sun ;  one  is  then  led  to  believe 
with  much  probability  that  the  planets  have  been 
formed  at  the  successive  limits  of  the  solar  atmosphere 
which,  contracting  by  the  cold,  ought  to  have  abandoned 
in  the  plane  of  its  equator  zones  of  vapors  which  the 
mutual  attraction  of  their  molecules  has  changed  into 


102      A   PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

clivers  spheroids.  The  satellites  have  been  similarly 
formed  by  the  atmospheres  of  their  respective  planets. 

I  have  developed  at  length  in  my  Exposition  of  the 
System  of  the  World  this  hypothesis,  which  appears  to 
me  to  satisfy  all  the  phenomena  which  this  system 
presents  us.  I  shall  content  myself  here  with  con- 
sidering that  the  angular  velocity  of  rotation  of  the 
sun  and  the  planets  being  accelerated  by  the  successive 
condensation  of  their  atmospheres  at  their  surfaces,  it 
ought  to  surpass  the  angular  velocity  of  revolution  of 
the  nearest  bodies  which  revolve  about  them.  Obser- 
vation has  indeed  confirmed  this  with  regard  to  the 
planets  and  satellites,  and  even  in  ratio  to  the  ring  of 
Saturn,  the  duration  of  whose  revolution  is  .438 
minutes,  while  the  duration  of  the  rotation  of  Saturn  is 
.427  minutes. 

In  this  hypothesis  the  comets  are  strangers  to  the 
planetary  system.  In  attaching  their  formation  to  that 
of  the  nebulae  they  may  be  regarded  as  small  nebulae 
at  the  nuclei,  wandering  from  systems  to  solar  systems, 
and  formed  by  the  condensation  of  the  nebulous  matter 
spread  out  in  such  great  profusion  in  the  universe. 
The  comets  would  be  thus,  in  relation  to  our  system,  as 
the  aerolites  are  relatively  to  the  Earth,  to  which  they 
would  appear  strangers.  When  these  stars  become 
visible  to  us  they  offer  so  perfect  resemblance  to  the 
nebulae  that  they  are  often  confounded  with  them ;  and 
it  is  only  by  their  movement,  or  by  the  knowledge  of 
all  the  nebulae  confined  to  that  part  of  the  heavens 
where  they  appear,  that  we  succeed  in  distinguishing 
them.  This  supposition  explains  in  a  happy  manner 
the  <jreat  extension  which  the  heads  and  tails  of  comets 


PROBABILITIES  AND  NATURAL  PHILOSOPHY.       103 

take  in  the  measure  that  they  approach  the  sun,  and  the 
extreme  rarity  of  these  tails  which,  in  spite  of  their 
immense  depth,  do  not  weaken  at  all  appreciably  the 
light  of  the  stars  which  we  look  across. 

When  the  little  nebulse  come  into  that  part  of  space 
where  the  attraction  of  the  sun  is  predominant,  and 
which  we  shall  call  the  sphere  of  activity  of  this  star, 
it  forces  them  to  describe  elliptic  or  hyperbolic  orbits. 
But  their  speed  being  equally  possible  in  all  directions 
they  ought  to  move  indifferently  in  all  the  senses  and 
under  all  inclinations  of  the  elliptic,  which  is  conform- 
able to  that  which  has  been  observed. 

The  great  eccentricity  of  the  cometary  orbits  results 
again  from  the  preceding  hypothesis.  Indeed  if  these 
orbits  are  elliptical  they  are  very  elongated,  since  their 
great  axes  are  at  least  equal  to  the  radius  of  the  sphere 
of  activity  of  the  sun.  But  these  orbits  may  be  hyper- 
bolic ;  and  if  the  axes  of  these  hyperbolae  are  not  very 
large  in  proportion  to  the  mean  distance  from  the  sun 
to  the  earth,  the  movement  of  the  comets  which  describe 
them  will  appear  sensibly  hyperbolic.  However,  of 
the  hundred  comets  of  which  we  already  have  the  ele- 
ments, not  one  has  appeared  certainly  to  move  in  an 
hyperbola;  it  is  necessary,  then,  that  the  chances  which 
give  an  appreciable  hyperbola  should  be  extremely 
rare  in  proportion  to  the  contrary  chances. 

The.  comets  are  so  small  that,  in  order  to  become 
visible,  their  perihelion  distance  ought  to  be  inconsider- 
able. Up  to  the  present  this  distance  has  surpassed 
only  twice  the  diameter  of  the  terrestrial  orbit,  and 
most  often  it  has  been  below  the  radius  of  this  orbit. 
It  is  conceived  that,  in  order  to  approach  so  near  the 


104      A  PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

sun,  their  speed  at  the  moment  of  their  entrance  into 
its  sphere  of  activity  ought  to  have  a  magnitude  and  a 
direction  confined  within  narrow  limits.  In  determin- 
ing by  the  analysis  of  probabilities  the  ratio  of  the 
chances  which,  in  these  limits,  give  an  appreciable 
hyperbola,  to  the  chances  which  give  an  orbit  which 
may  be  confounded  with  a  parabola,  I  have  found  that 
it  is  a  bet  of  at  least  6000  against  one  that  a  nebula 
which  penetrates  into  the  activity  of  the  sun  in  such  a 
manner  as  to  be  observed  will  describe  either  a  very 
elongated  ellipse  or  an  hyperbola.  By  the  magnitude 
of  its  axis,  the  latter  will  be  appreciably  confounded 
with  a  parabola  in  the  part  which  is  observed;  it  is 
then  not  surprising  that,  up  to  this  time,  hyperbolic 
movements  have  not  been  recognized. 

The  attraction  of  the  planets,  and,  perhaps  further,  the 
resistance  of  the  ethereal  centres,  ought  to  have  changed 
many  cometary  orbits  in  the  ellipses  whose  great  axis 
is  less  than  the  radius  of  the  sphere  of  activity  of  the 
sun,  which  augments  the  chances  of  the  elliptical  orbits. 
We  may  believe  that  this  change  has  taken  place  with 
the  comet  of  1759,  and  with  the  comet  whose  duration 
is  only  twelve  hundred  days,  and  which  will  reappear 
without  ceasing  in  this  short  interval,  unless  the 
evaporation  which  it  meets  at  each  of  its  returns  to  the 
perihelion  ends  by  rendering  it  invisible. 

We  are  able  further,  by  the  analysis  of  probabilities, 
to  verify  the  existence  or  the  influence  of  certain  causes 
whose  action  is  believed  to  exist  upon  organized  beings. 
Of  all  the  instruments  that  we  are  able  to  employ  in 
order  to  recognize  the  imperceptible  agents  of  nature 
the  most  sensitive  are  the  nerves,  especially  when  par- 


PROBABILITIES  AND  NATURAL   PHILOSOPHY.       105 

ticular  causes  increase  their  sensibility.  It  is  by  their 
aid  that  the  feeble  electricity  which  the  contact  of  two 
heterogeneous  metals  develops  has  been  discovered ; 
this  has  opened  a  vast  field  to  the  researches  of  physi- 
cists and  chemists.  The  singular  phenomena  which 
results  from  extreme  sensibility  of  the  nerves  in  some 
individuals  have  given  birth  to  divers  opinions  about  the 
existence  of  a  new  agent  which  has  been  named  animal 
magnetism,  about  the  action  on  ordinary  magnetism, 
and  about  the  influence  of  the  sun  and  moon  in  some 
nervous  affections,  and  finally,  about  the  impressions 
which  the  proximity  of  metals  or  of  running  water 
makes  felt.  It  is  natural  to  think  that  the  action  of 
these  causes  is  very  feeble,  and  that  it  may  be  easily 
disturbed  by  accidental  circumstances;  thus  because  in 
some  cases  it  is  not  manifested  at  all  its  existence 
ought  not  to  be  denied.  We  are  so  far  from  recogniz- 
ing all  the  agents  of  nature  and  their  divers  modes  of 
action  that  it  would  be  unphilosophical  to  deny  the 
phenomena  solely  because  they  are  inexplicable  in  the 
present  state  of  our  knowledge.  But  we  ought  to 
examine  them  with  an  attention  as  much  the  more 
scrupulous  as  it  appears  the  more  difficult  to  admit 
them  ;  and  it  is  here  that  the  calculation  of  probabilities 
becomes  indispensable  in  determining  to  just  what 
point  it  is  necessary  to  multiply  the  observations  or  the 
experiences  in  order  to  obtain  in  favor  of  the  agents 
which  they  indicate,  a  probability  superior  to  the 
reasons  which  can  be  obtained  elsewhere  for  not 
admitting  them. 

-  The  calculation  of  probabilities  can  make  appreciable 
the  advantages  and  the  inconveniences  of  the  methods 


io6      A  PHILOSOPHICAL   ESSAY   ON  PROBABILITIES. 

employed  in  the  speculative  sciences.  Thus  in  order 
to  recognize  the  best  of  the  treatments  in  use  in  the 
healing  of  a  malady,  it  is  sufficient  to  test  each  of  them 
on  an  equal  number  of  patients,  making  all  the  condi- 
tions exactly  similar;  the  superiority  of  the  most 
advantageous  treatment  will  manifest  itself  more  and 
more  in  the  measure  that  the  number  is  increased;  and 
the  calculation  will  make  apparent  the  corresponding 
probability  of  its  advantage  and  the  ratio  according  to 
which  it  is  superior  to  the  others. 


CHAPTER    X. 

APPLICATION   OF    THE    CALCULUS    OF    PROB- 
ABILITIES  TO   THE  MORAL  SCIENCES. 

WE  have  just  seen  the  advantages  of  the  analysis  of 
probabilities  in  the  investigation  of  the  laws  of  natural 
phenomena  whose  causes  are  unknown  or  so  compli- 
cated that  their  results  cannot  be  submitted  to  calculus. 
This  is  the  case  of  nearly  all  subjects  of  the  moral 
sciences.  So  many  unforeseen  causes,  either  hidden 
or  inappreciable,  influence  human  institutions  that  it  is 
impossible  to  judge  a  priori  the  results.  The  series  of 
events  which  time  brings  about  develops  these  results 
and  indicates  the  means  of  remedying  those  that  are 
harmful.  Wise  laws  have  often  been  made  in  this 
regard ;  but  because  we  had  neglected  to  conserve  the 
motives  many  have  been  abrogated  as  useless,  and  the 
fact  that  vexatious  experiences  have  made  the  need  felt 
anew  ought  to  have  reestablished  them. 

It  is  very  important  to  keep  in  each  branch  of  the 
public  administration  an  exact  register  of  the  results 
which  the  various  means  used  have  produced,  and  which 
are  so  many  experiences  made  on  a  large  scale  by 
governments.  Let  us  apply  to  the  political  and  moral 

107 


io8      A   PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

sciences  the  method  founded  upon  observation  and 
upon  calculus,  the  method  which  has  served  us  so  well 
in  the  natural  sciences.  Let  us  not  offer  in  the  least 
a  useless  and  often  dangerous  resistance  to  the 
inevitable  effects  of  the  progress  of  knowledge;  but  let 
us  change  only  with  an  extreme  circumspection  our 
institutions  and  the  usages  to  which  we  have  already 
so  long  conformed.  We  should  know  well  by  the 
experience  of  the  past  the  difficulties  which  they 
present ;  but  we  are  ignorant  of  the  extent  of  the  evils 
which  their  change  can  produce.  In  this  ignorance 
the  theory  of  probability  directs  us  to  avoid  all  change; 
especially  is  it  necessary  to  avoid  the  sudden  changes 
which  in  the  moral  world  as  well  as  in  the  physical 
world  never  operate  without  a  great  loss  of  vital  force. 
Already  the  calculus  of  probabilities  has  been  applied 
with  success  to  several  subjects  of  the  moral  sciences. 
I  shall  present  here  the  principal  results. 


CHAPTER   XL 

CONCERNING   THE  PROBABILITIES  OF  TESTI- 
MONIES. 

THE  majority  of  our  opinions  being  founded  on  the 
probability  of  proofs  it  is  indeed  important  to  submit  it 
to  calculus.  Things  it  is  true  often  become  impossible 
by  the  difficulty  of  appreciating  the  veracity  of  wit- 
nesses and  by  the  great  number  of  circumstances  which 
accompany  the  deeds  they  attest ;  but  one  is  able  in 
several  cases  to  resolve  the  problems  which  have  much 
analogy  with  the  questions  which  are  proposed  and 
whose  solutions  may  be  regarded  as  suitable  approxi- 
mations to  guide  and  to  defend  us  againt  the  errors  and 
the  dangers  of  false  reasoning  to  which  we  are  exposed. 
An  approximation  of  this  kind,  when  it  is  well  made, 
is  always  preferable  to  the  most  specious  reasonings. 
Let  us  try  then  to  give  some  general  rules  for  obtain- 
ing it. 

A  single  number  has  been  drawn  from  an  urn  which 
contains  a  thousand  of  them.  A  witness  to  this  draw- 
ing announces  that  number  79  is  drawn ;  one  asks  the 
probability  of  drawing  this  number.  Let  us  suppose 
that  experience  has  made  known  that  this  witness 

log 


no     A  PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 

deceives  one  time  in  ten,  so  that  the  probability  of  his 
testimony  is  TV  Here  the  event  observed  is  the  wit- 
ness attesting  that  number  79  is  drawn.  This  event 
may  result  from  the  two  following  hypotheses,  namely: 
that  the  witness  utters  the  truth  or  that  he  deceives. 
Following  the  principle  that  has  been  expounded  on 
the  probability  of  causes  drawn  from  events  observed 
it  is  necessary  first  to  determine  a  priori  the  probabil- 
ity of  the  event  in  each  hypothesis.  In  the  first,  the 
probability  that  the  witness  will  announce  number  79 
is  the  probability  itself  of  the  drawing  of  this  number, 
that  is  to  say,  TTTOTT-  It  is  necessary  to  multiply  it  by 
the  probability  j6ff  of  the  veracity  of  the  witness ;  one 
will  have  then  T¥|hn5  f°r  the  probability  of  the  event 
observed  in  this  hypothesis.  If  the  witness  deceives, 
number  79  is  not  drawn,  and  the  probability  of  this 
case  is  •$$$$.  But  to  announce  the  drawing  of  this 
number  the  witness  has  to  choose  it  among  the  999 
numbers  not  drawn ;  and  as  he  is  supposed  to  have  no 
motive  of  preference  for  the  ones  rather  than  the 
others,  the  probability  that  he  will  choose  number  79 
is  -577;  multiplying,  then,  this  probability  by  the  pre- 
ceding one,  we  shall  have  y^Vo  f°r  the  probability  that 
the  witness  will  announce  number  79  in  the  second 
hypothesis.  It  is  necessary  again  to  multiply  this 
probability  by  TV  of  the  hypothesis  itself,  which  gives 
uriinr  f°r  t^e  probability  of  the  event  relative  to  this 
hypothesis.  Now  if  we  form  a  fraction  whose  numera- 
tor is  the  probability  relative  to  the  first  hypothesis,  and 
whose  denominator  is  the  sum  of  the  probabilities  rela- 
tive to  the  two  hypotheses,  we  shall  have,  by  the  sixth 
principle,  the  probability  of  the  first  hypothesis,  and 


CONCERNING    THE  PROBABILITIES  OF  TESTIMONIES,   m 

this  probability  will  be  T9ff;  that  is  to  say,  the  veracity 
itself  of  the  witness.  This  is  likewise  the  probability 
of  the  drawing  of  number  79.  The  probability  of  the 
falsehood  of  the  witness  and  of  the  failure  of  drawing 
this  number  is  fa. 

If  the  witness,  wishing  to  deceive,  has  some  interest 
in  choosing  number  79  among  the  numbers  not  drawn, 
— if  he  judges,  for  example,  that  having  placed  upon 
this  number  a  considerable  stake,  the  announcement 
of  its  drawing  will  increase  his  credit,  the  probability 
that  he  will  choose  this  number  will  no  longer  be  as 
at  first,  -jfg,  it  will  then  be  £,  £,  etc.,  according  to  the 
interest  that  he  will  have  in  announcing  its  drawing. 
Supposing  it  to  be  |,  it  will  be  necessary  to  multiply 
by  this  fraction  the  probability  TVVo  m  order  to  get  in 
the  hypothesis  of  the  falsehood  the  probability  of  the 
event  observed,  which  it  is  necessary  still  to  multiply 
by  y^,  which  gives  TihhjT  f°r  the  probability  of  the 
event  in  the  second  hypothesis.  Then  the  probability 
of  the  first  hypothesis,  or  of  the  drawing  of  number  79, 
is  reduced  by  the  preceding  rule  to  yfg-.  It  is  then 
very  much  decreased  by  the  consideration  of  the  in- 
terest which  the  witness  may  have  in  announcing  the 
drawing  of  number  79.  In  truth  this  same  interest 
increases  the  probability  -^  that  the  witness  will  speak 
the  truth  if  number  79  is  drawn.  But  this  probability 
cannot  exceed  unity  or  |£ ;  thus  the  probability  of  the 
drawing  of  number  79  will  not  surpass  Ty>T-  Common 
sense  tells  us  that  this  interest  ought  to  inspire  distrust, 
but  calculus  appreciates  the  influence  of  it. 

The  probability  a  priori  of  the  number  announced 
by  the  witness  is  unity  divided  by  the  number  of  the 


H2      A   PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

numbers  in  the  urn;  it  is  changed  by  virtue  of  the 
proof  into  the  veracity  itself  of  the  witness;  it  may  then 
be  decreased  by  the  proof.  If,  for  example,  the  urn 
contains  only  two  numbers,  which  gives  £  for  the 
probability  a  priori  of  the  drawing  of  number  I ,  and  if 
the  veracity  of  a  witness  who  announces  it  is  T%,  this 
drawing  becomes  less  probable.  Indeed  it  is  apparent, 
since  the  witness  has  then  more  inclination  towards  a 
falsehood  than  towards  the  truth,  that  his  testimony 
ought  to  decrease  the  probability  of  the  fact  attested 
every  time  that  this  probability  equals  or  surpasses  £. 
But  if  there  are  three  numbers  in  the  urn  the  probability 
a  priori  of  the  drawing  of  number  I  is  increased  by 
the  affirmation  of  a  witness  whose  veracity  surpasses  £. 
Suppose  now  that  the  urn  contains  999  black  balls 
and  one  white  ball,  and  that  one  ball  having  been 
drawn  a  witness  of  the  drawing  announces  that  this 
ball  is  white.  The  probability  of  the  event  observed, 
determined  a  priori  in  the  first  hypothesis,  will  be  here, 
as  in  the  preceding  question,  equal  to  -foooir-  But  m 
the  hypothesis  where  the  witness  deceives,  the  white 
ball  is  not  drawn  and  the  probability  of  this  case 
is  TV(TV  It  ls  necessary  to  multiply  it  by  the  prob- 
ability TV  of  the  falsehood,  which  gives  T|||^  for  the 
probability  of  the  event  observed  relative  to  the  second 
hypothesis.  This  probability  was  only  Tol7nr  m  tne 
preceding  question;  this  great  difference  results  from 
this — that  a  black  ball  having  been  drawn  the  witness 
who  wishes  to  deceive  has  no  choice  at  all  to  make 
among  the  999  balls  not  drawn  in  order  to  announce 
the  drawing  of  a  white  ball.  Now  if  one  forms  two 
fractions  whose  numerators  are  the  probabilities  relative 


CONCERNING    THE  PROBABILITIES  OF  TESTIMONIES.    113 

to  each  hypothesis,  and  whose  common  denominator  is 
the  sum  of  these  probabilities,  one  will  have  i^^  for 
the  probability  of  the  first  hypothesis  and  of  the  drawing 
of  a  white  ball,  and  TVir9ff  f°r  the  probability  of  the 
second  hypothesis  and  of  the  drawing  of  a  black  ball. 
This  last  probability  strongly  approaches  certainty ;  it 
would  approach  it  much  nearer  and  would  become 
TVoVoVs  if  the  urn  contained  a  million  balls  of  which 
one  was  white,  the  drawing  of  a  white  ball  becoming 
then  much  more  extraordinary.  We  see  thus  how  the 
probability  of  the  falsehood  increases  in  the  measure 
that  the  deed  becomes  more  extraordinary. 

We  have  supposed  up  to  this  time  that  the  witness 
was  not  mistaken  at  all ;  but  if  one  admits,  however, 
the  chance  of  his  error  the  extraordinary  incident 
becomes  more  improbable.  Then  in  place  of  the  two 
hypotheses  one  will  have  the  four  following  ones, 
namely:  that  of  the  witness  not  deceiving  and  not  being 
mistaken  at  all ;  that  of  the  witness  not  deceiving  at 
all  and  being  mistaken ;  the  hypothesis  of  the  witness 
deceiving  and  not  being  mistaken  at  all;  finally,  that 
of  the  witness  deceiving  and  being  mistaken.  Deter- 
mining a  priori  in  each  of  these  hypotheses  the  prob- 
ability of  the  event  observed,  we  find  by  the  sixth 
principle  the  probability  that  the  fact  attested  is  false 
equal  to  a  fraction  whose  numerator  is  the  number  of 
black  balls  in  the  urn  multiplied  by  the  sum  of  the 
probabilities  that  the  witness  does  not  deceive  at  all 
and  is  mistaken,  or  that  he  deceives  and  is  not  mis- 
taken, and  whose  denominator  is  this  numerator 
augmented  by  the  sum  of  the  probabilities  that  the 
witness  does  not  deceive  at  all  and  is  not  mistaken  at 


H4      A  PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 

all,  or  that  he  deceives  and  is  mistaken  at  the  same 
time.  We  see  by  this  that  if  the  number  of  black 
balls  in  the  urn  is  very  great,  which  renders  the  draw- 
ing of  the  white  ball  extraordinary,  the  probability  that 
the  fact  attested  is  not  true  approaches  most  nearly  to 
certainty. 

Applying  this  conclusion  to  all  extraordinary  deeds 
it  results  from  it  that  the  probability  of  the  error  or  of 
the  falsehood  of  the  witness  becomes  as  much  greater 
as  the  fact  attested  is  more  extraordinary.  Some 
authors  have  advanced  the  contrary  on  this  basis  that 
the  view  of  an  extraordinary  fact  being  perfectly  similar 
to  that  of  an  ordinary  fact  the  same  motives  ought  to 
lead  us  to  give  the  witness  the  same  credence  when  he 
affirms  the  one  or  the  other  of  these  facts.  Simple 
common  sense  rejects  such  a  strange  assertion ;  but  the 
calculus  of  probabilities,  while  confirming  the  findings 
of  common  sense,  appreciates  the  greatest  improbability 
of  testimonies  in  regard  to  extraordinary  facts. 

These  authors  insist  and  suppose  two  witnesses 
equally  worthy  of  belief,  of  whom  the  first  attests  that 
he  saw  an  individual  dead  fifteen  days  ago  whom  the 
second  witness  affirms  to  have  seen  yesterday  full 
of  life.  The  one  or  the  other  of  these  facts  offers  no 
improbability.  The  reservation  of  the  individual  is  a 
result  of  their  combination ;  but  the  testimonies  do  not 
bring  us  at  all  directly  to  this  result,  although  the 
credence  which  is  due  these  testimonies  ought  not  to 
be  decreased  by  the  fact  that  the  result  of  their  com- 
bination is  extraordinary. 

But  if  the  conclusion  which  results  from  the  com- 
bination of  the  testimonies  was  impossible  one  of  them 


CONCERNING    THE  PROBABILITIES   OF   TESTIMONIES.    115 

would  be  necessarily  false;  but  an  impossible  conclu- 
sion is  the  limit  of  extraordinary  conclusions,  as  error 
is  the  limit  of  improbable  conclusions;  the  value  of  the 
testimonies  which  becomes  zero  in  the  case  of  an 
impossible  conclusion  ought  then  to  be  very  much 
decreased  in  that  of  an  extraordinary  conclusion. 
This  is  indeed  confirmed  by  the  calculus  of  prob- 
abilities. 

In  order  to  make  it  plain  let  us  consider  two  urns,  A 
and  B,  of  which  the  first  contains  a  million  white  balls 
and  the  second  a  million  black  balls.  One  draws  from 
one  of  these  urns  a  ball,  which  he  puts  back  into  the 
other  urn,  from  which  one  then  draws  a  ball.  Two 
witnesses,  the  one  of  the  first  drawing,  the  other  of  the 
second,  attest  that  the  ball  which  they  have  seen  drawn 
is  white  without  indicating  the  urn  from  which  it  has 
been  drawn.  Each  testimony  taken  alone  is  not 
improbable;  and  it  is  easy  to  see  that  the  probability 
of  the  fact  attested  is  the  veracity  itself  of  the  witness. 
But  it  follows  from  the  combination  of  the  testimonies 
that  a  white  ball  has  been  extracted  from  the  urn  A  at 
the  first  draw,  and  that  then  placed  in  the  urn  B  it 
has  reappeared  at  the  second  draw,  which  is  very 
extraordinary;  for  this  second  urn,  containing  then  one 
white  ball  among  a  million  black  balls,  the  probability 
of  drawing  the  white  ball  is  yc-UFor-  ^n  order  to 
determine  the  diminution  which  results  in  the  prob- 
ability of  the  thing  announced  by  the  two  witnesses 
we  shall  notice  that  the  event  observed  is  here  the 
affirmation  by  each  of  them  that  the  ball  which  he  has 
seen  extracted  is  white.  Let  us  represent  by  T9T  the 
probability  that  he  announces  the  truth,  which  can 


Ii6      A  PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

occur  in  the  present  case  when  the  witness  does  not 
deceive  and'  is  not  mistaken  at  all,  and  when  he 
deceives  and  is  mistaken  at  the  same  time.  One  may 
form  the  four  following  hypotheses : 

1st.  The  first  and  second  witness  speak  the  truth. 
Then  a  white  ball  has  at  first  been  drawn  from  the  urn 
A,  and  Ihe  probability  of  this  event  is  |,  since  the  ball 
drawn  al  the  first  draw  may  have  been  drawn  either 
from  the  one  or  the  other  urn.  Consequently  the  ball 
drawn,  placed  in  the  urn  B,  has  reappeared  at  the 
second  draw;  the  probability  of  this  event  is 
the  probability  of  the  fact  announced  is  then 
Multiplying  it  by  the  product  of  the  probabilities  -fa 
and  y9^  that  the  witnesses  speak  the  truth  one  will 
have  ^nnrVoinr  f°r  the  probability  of  the  event  ob- 
served in  this  first  hypothesis. 

2d.  The  first  witness  speaks  the  truth  and  the  second 
does  not,  whether  he  deceives  and  is  not  mistaken  or 
he  does  not  deceive  and  is  mistaken.  Then  a  white 
ball  has  been  drawn  from  the  urn  A  at  the  first  draw, 
and  the  probability  of  this  event  is  £.  Then  this  ball 
having  been  placed  in  the  urn  B  a  black  ball  has  been 
drawn  from  it:  the  probability  of  such  drawing  is 
|£_o_o  0.0.0 .  one  has  then  £#{$£!$  for  the  probability  of 
the  compound  event.  Multiplying  it  by  the  product 
of  the  two  probabilities  T9¥  and  TV  that  the  first  witness 
speaks  the  truth  and  that  the  second  does  not,  one 
will  have  y^^fjfo.  for  the  probability  for  the  event 
observed  in  the  second  hypothesis. 

3d.  The  first  witness  does  not  speak  the  truth  and 
the  second  announces  it.  Then  a  black  ball  has  been 
drawn  from  the  urn  B  at  the  first  drawing,  and  after 


CONCERNING    THE  PROBABILITIES  OF  TESTIMONIES,   n? 

having  been  placed  in  the  urn  A  a  white  ball  has  been 
drawn  from  this  urn.  The  probability  of  the  first  of 
these  events  is  £  and  that  of  the  second  is  T£#$S$T ;  the 
probability  of  the  compound  event  is  then  i^^-^f. 
Multiplying  it  by  the  product  of  the  probabilities  13jr 
and  yV  that  the  first  witness  does  not  speak  the  truth 
and  that  the  second  announces  it,  one  will  have 
siiHfTmHta  for  tne  probability  of  the  event  observed 
relative  to  this  hypothesis. 

4th.  Finally,  neither  of  the  witnesses  speaks  the  truth. 
Then  a  black  ball  has  been  drawn  from  the  urn  B  at 
the  first  draw;  then  having  been  placed  in  the  urn  A 
it  has  reappeared  at  the  second  drawing:  the  prob- 
ability of  this  compound  event  is  aooooo?-  Multiply- 
ing it  by  the  product  of  the  probabilities  -fa  and  y1^-  that 
each  witness  does  not  speak  the  truth  one  will  have 
200000200  f°r  tne  probability  of  the  event  observed  in 
this  hypothesis. 

Now  in  order  to  obtain  the  probability  of  the  thing 
announced  by  the  two  witnesses,  namely,  that  a  white 
ball  has  been  drawn  at  each  draw,  it  is  necessary  to 
divide  the  probability  corresponding  to  the  first  hy- 
pothesis by  the  sum  of  the  probabilities  relative  to 
the  four  hypotheses ;  and  then  one  has  for  this  prob- 
ability y-g-ooooas'  an  extremely  small  fraction. 

If  the  two  witnesses  affirm  the  first,  that  a  white 
ball  has  been  drawn  from  one  of  the  two  urns  A  and 
B;  the  second  that  a  white  ball  has  been  likewise 
drawn  from  one  of  the  two  urns  A'  and  B',  quite 
similar  to  the  first  ones,  the  probability  of  the  thing 
announced  by  the  two  witnesses  will  be  the  product  of 
the  probabilities  of  their  testimonies,  or  y\V;  it  will  then 


n8      A  PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

be  at  least  a  hundred  and  eighty  thousand  times 
greater  than  the  preceding  one.  One  sees  by  this  how 
much,  in  the  first  case,  the  reappearance  at  the  second 
draw  of  the  white  ball  drawn  at  the  first  draw,  the 
extraordinary  conclusion  of  the  two  testimonies  de- 
creases the  value  of  it. 

We  would  give  no  credence  to  the  testimony  of  a 
man  who  should  attest  to  us  that  in  throwing  a  hundred 
dice  into  the  air  they  had  all  fallen  on  the  same  face. 
If  we  had  ourselves  been  spectators  of  this  event  we 
should  believe  our  own  eyes  only  after  having  carefully 
examined  all  the  circumstances,  and  after  having 
brought  in  the  testimonies  of  other  eyes  in  order  to  be 
quite  sure  that  there  had  been  neither  hallucination  nor 
deception.  But  after  this  examination  we  should  not 
hesitate  to  admit  it  in  spite  of  its  extreme  improbability; 
and  no  one  would  be  tempted,  in  order  to  explain  it,  to 
recur  to  a  denial  of  the  laws  of  vision.  We  ought  to 
conclude  from  it  that  the  probability  of  the  constancy 
of  the  laws  of  nature  is  for  us  greater  than  this,  that 
the  event  in  question  has  not  taken  place  at  all — a 
probability  greater  than  that  of  the  majority  of  his- 
torical facts  which  we  regard  as  incontestable.  One 
may  judge  by  this  the  immense  weight  of  testimonies 
necessary  to  admit  a  suspension  of  natural  laws,  and 
how  improper  it  would  be  to  apply  to  this  case  the 
ordinary  rules  of  criticism.  All  those  who  without 
offering  this  immensity  of  testimonies  support  this 
when  making  recitals  of  events  contrary  to  those  laws, 
decrease  rather  than  augment  the  belief  which  they 
wish  to  inspire ;  for  then  those  recitals  render  very 
probable  the  error  or  the  falsehood  of  their  authors. 


CONCERNING   THE  PROBABILITIES  OF  TESTIMONIES.   119 

But  that  which  diminishes  the  belief  of  educated  men 
increases  often  that  of  the  uneducated,  always  greedy 
for  the  wonderful. 

There  are  things  so  extraordinary  that  nothing  can 
balance  their  improbability.  But  this,  by  the  effect  of 
a  dominant  opinion,  can  be  weakened  to  the  point  of 
appearing  inferior  to  the  probability  of  the  testimonies ; 
and  when  this  opinion  changes  an  absurd  statement 
admitted  unanimously  in  the  century  which  has  given 
it  birth  offers  to  the  following  centuries  only  a  new 
proof  of  the  extreme  influence  of  the  general  opinion 
upon  the  more  enlightened  minds.  Two  great  men  of 
the  century  of  Louis  XIV. — Racine  and  Pascal — are 
striking  examples  of  this.  It  is  painful  to  see  with 
what  complaisance  Racine,  this  admirable  painter  of 
the  human  heart  and  the  most  perfect  poet  that  has 
ever  lived,  reports  as  miraculous  the  recovery  of  Mile. 
Perrier,  a  niece  of  Pascal  and  a  day  pupil  at  the 
monastery  of  Port-Royal;  it  is  painful  to  read  the 
reasons  by  which  Pascal  seeks  to  prove  that  this  miracle 
should  be  necessary  to  religion  in  order  to  justify  the 
doctrine  of  the  monks  of  this  abbey,  at  that  time  perse- 
cuted by  the  Jesuits.  The  young  Perrier  had  been 
afflicted  for  three  years  and  a  half  by  a  lachrymal  fistula; 
she  touched  her  afflicted  eye  with  a  relic  which  was 
pretended  to  be  one  of  the  thorns  of  the  crown  of  the 
Saviour  and  she  had  faith  in  instant  recovery.  Some 
days  afterward  the  physicians  and  the  surgeons  attest 
the  recovery,  and  they  declare  that  nature  and  the 
remedies  have  had  no  part  in  it.  This  event,  which 
took  place  in  1656,  made  a  great  sensation,  and  "all 
Paris  rushed,"  says  Racine,  "to  Port-Royal.  The 


120      A  PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

crowd  increased  from  day  to  day,  and  God  himself 
seemed  to  take  pleasure  in  authorizing  the  devotion  of 
the  people  by  the  number  of  miracles  which  were  per- 
formed in  this  church."  At  this  time  miracles  and 
sorcery  did  not  yet  appear  improbable,  and  one  did  not 
hesitate  at  all  to  attribute  to  them  the  singularities  of 
nature  which  could  not  be  explained  otherwise. 

This  manner  of  viewing  extraordinary  results  is 
found  in  the  most  remarkable  works  of  the  century  of 
Louis  XIV. ;  even  in  the  Essay  on  the  Human  Under- 
standing by  the  philosopher  Locke,  who  says,  in 
speaking  of  the  degree  of  assent:  "  Though  the  com- 
mon experience  and  the  ordinary  course  of  things  have 
justly  a  mighty  influence  on  the  minds  of  men,  to  make 
them  give  or  refuse  credit  to  anything  proposed  to  their 
belief;  yet  there  is  one  case,  wherein  the  strangeness 
of  the  lact  lessens  not  the  assent  to  a  fair  testimony  of  it. 
For  where  such  supernatural  events  are  suitable  to  ends 
aimed  at  by  him  who  has  the  power  to  change  the 
course  of  nature,  there,  under  such  circumstances,  they 
maybe  the  fitter  to  procure  belief,  by  how  much  the  more 
they  are  beyond  or  contrary  to  ordinary  observation. " 
The  true  principles  of  the  probability  of  testimonies 
having  been  thus  misunderstood  by  philosophers  to 
whom  reason  is  principally  indebted  for  its  progress,  I 
have  thought  it  necessary  to  present  at  length  the 
results  of  calculus  upon  this  important  subject. 

There  comes  up  naturally  at  this  point  the  discussion 
of  a  famous  argument  of  Pascal,  that  Craig,  an  English 
mathematician,  has  produced  under  a  geometric  form. 
Witnesses  declare  that  they  have  it  from  Divinity  that 
in  conforming  to  a  certain  thing  one  will  enjoy  not  one 


CONCERNING    THE  PROBABILITIES   OF   TESTIMONIES.   121 

or  two  but  an  infinity  of  happy  lives.  However  feeble 
the  probability  of  the  proofs  may  be,  provided  that  it 
be  not  infinitely  small,  it  is  clear  that  the  advantage  of 
those  who  conform  to  the  prescribed  thing  is  infinite 
since  it  is  the  product  of  this  probability  and  an  infinite 
good ;  one  ought  not  to  hesitate  then  to  procure  for 
oneself  this  advantage. 

This  argument  is  based  upon  the  infinite  number  of 
happy  lives  promised  in  the  name  of  the  Divinity  by 
the  witnesses;  it  is  necessary  then  to  prescribe  them, 
precisely  because  they  exaggerate  their  promises 
beyond  all  limits,  a  consequence  which  is  repugnant  to 
good  sense.  Also  calculus  teaches  us  that  this 
exaggeration  itself  enfeebles  the  probability  of  their 
testimony  to  the  point  of  rendering  it  infinitely  small 
or  zero.  Indeed  this  case  is  similar  to  that  of  a  witness 
who  should  announce  the  drawing  of  the  highest 
number  from  an  urn  filled  with  a  great  number  ot 
numbers,  one  of  which  has  been  drawn  and  who  would 
have  a  great  interest  in  announcing  the  drawing  of  this 
number.  One  has  already  seen  how  much  this  interest 
enfeebles  his  testimony.  In  evaluating  only  at  £  the 
probability  that  if  the  witness  deceives  he  will  choose 
the  largest  number,  calculus  gives  the  probability  of 
his  announcement  as  smaller  than  a  fraction  whose 
numerator  is  unity  and  whose  denominator  is  unity 
plus  the  half  of  the  product  of  the  number  of  the  num- 
bers by  the  probability  of  falsehood  considered  a  priori 
or  independently  of  the  announcement.  In  order  to 
compare  this  case  to  that  of  the  argument  of  Pascal  it 
is  sufficient  to  represent  by  the  numbers  in  the  urn  all 
the  possible  numbers  of  happy  lives  which  the  number 


122      A  PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

of  these  numbers  renders  infinite ;  and  to  observe  that 
if  the  witnesses  deceive  they  have  the  greatest  interest, 
in  order  to  accredit  their  falsehood,  in  promising  an 
eternity  of  happiness.  The  expression  of  the  prob- 
ability of  their  testimony  becomes  then  infinitely  small. 
Multiplying  it  by  the  infinite  number  of  happy  lives 
promised,  infinity  would  disappear  from  the  product 
which  expresses  the  advantage  resultant  from  this 
promise  which  destroys  the  argument  of  Pascal. 

Let  us  consider  now  the  probability  of  the  totality 
of  several  testimonies  upon  an  established  fact.  In 
order  to  fix  our  ideas  let  us  suppose  that  the  fact  be 
the  drawing  of  a  number  from  an  urn  which  contains  a 
hundred  of  them,  and  of  which  one  single  number  has 
been  drawn.  Two  witnesses  of  this  drawing  announce 
that  number  2  has  been  drawn,  and  one  asks  for  the 
resultant  probability  of  the  totality  of  these  testimonies. 
One  may  form  these  two  hypotheses:  the  witnesses 
speak  the  truth;  the  witnesses  deceive.  In  the  first 
hypothesis  the  number  2  is  drawn  and  the  probability 
of  this  event  is  -j-J-j-.  It  is  necessary  to  multiply  it  by 
the  product  of  the  veracities  of  the  witnesses,  veracities 
which  we  will  suppose  to  be  T97  and  T\:  one  will  have 
then  T^VTFIT  for  the  probability  of  the  event  observed  in 
this  hypothesis.  In  the  second,  the  number  2  is  not 
drawn  and  the  probability  of  this  event  is  y9^.  But 
the  agreement  of  the  witnesses  requires  then  that  in 
seeking  to  deceive  they  both  choose  the  number  2  from 
the  99  numbers  not  drawn:  the  probability  of  this 
choice  if  the  witnesses  do  not  have  a  secret  agreement 
is  the  product  of  the  fraction  5\  by  itself;  it  becomes 
necessary  then  to  multiply  these  two  probabilities 


CONCERNING  THE  PROBABILITIES  OF   TESTIMONIES.   123 

together,  and  by  the  product  of  the  probabilities  y1^  and 
Y3^  that  the  witnesses  deceive;  one  will  have  thus 
sygVuir  f°r  the  probability  of  the  event  observed  in  the 
second  hypothesis.  Now  one  will  have  the  probability 
of  the  fact  attested  or  of  the  drawing  of  number  2  in 
dividing  the  probability  relative  to  the  first  hypothesis 
by  the  sum  of  the  probabilities  relative  to  the  two 
hypotheses ;  this  probability  will  be  then  f$|-|,  and  the 
probability  of  the  failure  to  draw  this  number  and  of 
the  falsehood  of  the  witnesses  will  be  ^ViF' 

If  the  urn  should  contain  only  the  numbers  I  and  2 
one  would  find  in  the  same  manner  f  £  for  the  prob- 
ability of  the  drawing  of  number  2,  and  consequently 
^  for  the  probability  of  the  falsehood  of  the  witnesses, 
a  probability  at  least  ninety-four  times  larger  than  the 
preceding  one.  One  sees  by  this  how  much  the  prob- 
ability of  the  falsehood  of  the  witnesses  diminishes 
when  the  fact  which  they  attest  is  less  probable  in 
itself.  Indeed  one  conceives  that  then  the  accord  of 
the  witnesses,  when  they  deceive,  becomes  more  diffi- 
cult, at  least  when  they  do  not  have  a  secret  agree- 
ment, which  we  do  not  suppose  here  at  all. 

In  the  preceding  case  where  the  urn  contained  only 
two  numbers  the  a  priori  probability  of  the  fact  attested 
is  ^,  the  resultant  probability  of  the  testimonies  is  the 
product  of  the  veracities  of  the  witnesses  divided  by 
this  product  added  to  that  of  the  respective  probabilities 
of  their  falsehood. 

It  now  remains  for  us  to  consider  the  influence  of 
time  upon  the  probability  of  facts  transmitted  by  a 
traditional  chain  of  witnesses.  It  is  clear  that  this 
probability  ought  to  diminish  in  proportion  as  the  chain 


124      A  PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

is  prolonged.  If  the  fact  has  no  probability  itself,  such 
as  the  drawing  of  a  number  from  an  urn  which  contains 
an  infinity  of  them,  that  which  it  acquires  by  the  testi- 
monies decreases  according  to  the  continued  product 
of  the  veracity  of  the  witnesses.  If  the  fact  has  a 
probability  in  itself;  if,  for  example,  this  fact  is  the 
drawing  of  the  number  2  from  an  urn  which  contains 
an  infinity  of  them,  and  of  which  it  is  certain  that  one 
has  drawn  a  single  number;  that  which  the  traditional 
chain  adds  to  this  probability  decreases,  following  a 
continued  product  of  which  the  first  factor  is  the  ratio 
of  the  number  of  numbers  in  the  urn  less  one  to  the 
same  number,  and  of  which  each  other  factor  is  the 
veracity  of  each  witness  diminished  by  the  ratio  or"  the 
probability  of  his  falsehood  to  the  number  of  the  num- 
bers in  the  urn  less  one;  so  that  the  limit  of  the  prob- 
ability of  the  fact  is  that  of  this  fact  considered  a  priori, 
or  independently  of  the  testimonies,  a  probability  equal 
to  unity  divided  by  the  number  of  the  numbers  in  the 
urn. 

The  action  of  time  enfeebles  then,  without  ceasing, 
the  probability  of  historical  facts  just  as  it  changes  the 
most  durable  monuments.  One  can  indeed  diminish 
it  by  multiplying  and  conserving  the  testimonies  and 
the  monuments  which  support  them.  Printing  offers 
for  this  purpose  a  great  means,  unfortunately  unknown 
to  the  ancients.  In  spite  of  the  infinite  advantages 
which  it  procures  the  physical  and  moral  revolutions 
by  which  the  surface  of  this  globe  will  always  be 
agitated  will  end,  in  conjunction  with  the  inevitable 
effect  of  time,  by  rendering  doubtful  after  thousands  of 


CONCERNING  THE  PROBABILITIES   OF  TESTIMONIES.    125 

years  the  historical  facts  regarded  to-day  as  the  most 
certain. 

Craig  has  tried  to  submit  to  calculus  the  gradual 
enfeebling  of  the  proofs  of  the  Christian  religion ;  sup- 
posing that  the  world  ought  to  end  at  the  epoch  when 
it  will  cease  to  be  probable,  he  finds  that  this  ought  to 
take  place  1454  years  after  the  time  when  he  writes. 
But  his  analysis  is  as  faulty  as  his  hypothesis  upon  the 
duration  of  the  moon  is  bizarre. 


CHAPTER   XII. 

CONCERNING   THE  SELECTIONS  AND   THE 
DECISIONS  OF  ASSEMBLIES. 

THE  probability  of  the  decisions  of  an  assembly 
depends  upon  the  plurality  of  votes,  the  intelligence 
and  the  impartiality  of  the  members  who  compose  it. 
So  many  passions  and  particular  interests  so  often  add 
their  influence  that  it  is  impossible  to  submit  this  prob- 
ability to  calculus.  There  are,  however,  some  general 
results  dictated  by  simple  common  sense  and  confirmed 
by  calculus.  If,  for  example,  the  assembly  is  poorly 
informed  about  the  subject  submitted  to  its  decision,  if 
this  subject  requires  delicate  considerations,  or  if  the 
truth  on  this  point  is  contrary  to  established  prejudices, 
so  that  it  would  be  a  bet  of  more  than  one  against  one 
that  each  voter  will  err;  then  the  decision  of  the 
majority  will  be  probably  wrong,  and  the  fear  of  it  will 
be  the  better  based  as  the  assembly  is  more  numerous. 
It  is  important  then,  in  public  affairs,  that  assemblies 
should  have  to  pass  upon  subjects  within  reach  of  the 
greatest  number ;  it  is  important  for  them  that  informa- 
tion be  generally  diffused  and  that  good  works  founded 
upon  reason  and  experience  should  enlighten  those 

126 


SELECTIONS  AND  DECISIONS   OF  ASSEMBLIES.      127 

who  are  called  to  decide  the  lot  of  their  fellows  or  to 
govern  them,  and  should  forewarn  them  against  false 
ideas  and  the  prejudices  of  ignorance.  Scholars  have 
had  frequent  occasion  to  remark  that  first  conceptions 
often  deceive  and  that  the  truth  is  not  always  probable. 

It  is  difficult  to  understand  and  to  define  the  desire 
of  an  assembly  in  the  midst  of  a  variety  of  opinions  of 
its  members.  Let  us  attempt  to  give  some  rules  in 
regard  to  this  matter  by  considering  the  two  most 
ordinary  cases :  the  election  among  several  candidates, 
and  that  among  several  propositions  relative  to  the 
same  subject. 

When  an  assembly  has  to  choose  among  several 
candidates  who  present  themselves  for  one  or  for  several 
places  of  the  same  kind,  that  which  appears  simplest 
is  to  have  each  voter  write  upon  a  ticket  the  names  of 
all  the  candidates  according  to  the  order  of  merit  that 
he  attributes  to  them.  Supposing  that  he  classifies 
them  in  good  faith,  the  inspection  of  these  tickets  will 
give  the  results  of  the  elections  in  such  a  manner  that 
the  candidates  may  be  compared  among  themselves; 
so  that  new  elections  can  give  nothing  more  in  this 
regard.  It  is  a  question  now  to  conclude  the  order  of 
preference  which  the  tickets  establish  among  the  candi- 
dates. Let  us  imagine  that  one  gives  to  each  voter  an 
urn  which  contains  an  infinity  of  balls  by  means  of 
which  he  is  able  to  shade  all  the  degrees  of  merit  of 
the  candidates ;  let  us  conceive  again  that  he  draws 
from  his  urn  a  number  of  balls  proportional  to  the 
merit  of  each  candidate,  and  let  us  suppose  this  number 
written  upon  a  ticket  at  the  side  of  the  name  of  the 
candidate.  It  is  clear  that  by  making  a  sum  of  all  the 


128      A  PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 

numbers  relative  to  each  candidate  upon  each  ticket, 
that  one  of  all  the  candidates  who  shall  have  the 
largest  sum  will  be  the  candidate  whom  the  assembly 
prefers;  and  that  in  general  the  order  of  preference  of 
the  candidates  will  be  that  of  the  sums  relative  to  each 
of  them.  But  the  tickets  do  not  mark  at  all  the  num- 
ber of  balls  which  each  voter  gives  to  the  candidates ; 
they  indicate  solely  that  the  first  has  more  of  them  than 
the  second,  the  second  more  than  the  third,  and  so  on. 
In  supposing  then  at  first  upon  a  given  ticket  a  certain 
number  of  balls  all  the  combinations  of  the  inferior 
numbers  which  fulfil  the  preceding  conditions  are 
equally  admissible;  and  one  will  have  the  number  of 
balls  relative  to  each  candidate  by  making  a  sum  of  all 
the  numbers  which  each  combination  gives  him  and 
dividing  it  by  the  entire  number  of  combinations.  A 
very  simple  analysis  shows  that  the  numbers  which 
must  be  written  upon  each  ticket  at  the  side  of  the  last 
name,  of  the  one  before  the  last,  etc.,  are  proportional 
to  the  terms  of  the  arithmetical  progression  I,  2,  3, 
etc.  Writing  then  thus  upon  each  ticket  the  terms  of 
this  progression,  and  adding  the  terms  relative  to  each 
candidate  upon  these  tickets,  the  divers  sums  will  indi- 
cate by  their  magnitude  the  order  of  their  preference 
which  ought  to  be  established  among  the  candidates. 
Such  is  the  mode  of  election  which  The  Theory  of 
Probabilities  indicates.  Without  doubt  it  would  be 
better  if  each  voter  should  write  upon  his  ticket  the 
names  of  the  candidates  in  the  order  of  merit  which  he 
attributes  to  them.  But  particular  interests  and  many 
strange  considerations  of  merit  would  affect  this  order 
and  place  sometimes  in  the  last  rank  the  candidate 


SELECTIONS  AND  DECISIONS  OF  ASSEMBLIES.      129 

most  formidable  to  that  one  whom  one  prefers,  which 
gives  too  great  an  advantage  to  the  candidates  of 
mediocre  merit.  Likewise  experience  has  caused  the 
abandonment  of  this  mode  of  election  in  the  societies 
which  had  adopted  it. 

The  election  by  the  absolute  majority  of  the  suffrages 
unites  to  the  certainty  of  not  admitting  any  one  of  the 
candidates  whom  this  majority  rejects,  the  advantage 
of  expressing  most  often  the  desire  of  the  assembly. 
It  always  coincides  with  the  preceding  mode  when 
there  are  only  two  candidates.  Indeed  it  exposes  an 
assembly  to  the  inconvenience  of  rendering  elections 
interminable.  But  experience  has  shown  that  this 
inconvenience  is  nil,  and  that  the  general  desire  to  put 
an  end  to  elections  soon  unites  the  majority  of  the 
suffrages  upon  one  of  the  candidates. 

The  choice  among  several  propositions  relative  to 
the  same  object  ought  to  be  subjected,  seemingly,  to 
the  same  rules  as  the  election  among  several  candi- 
dates. But  there  exists  between  the  two  cases  this 
difference,  namely,  that  the  merit  of  a  candidate  does 
not  exclude  that  of  his  competitors;  but  if  it  is  neces- 
sary to  choose  among  propositions  which-  are  contrary, 
the  truth  of  the  one  excludes  the  truth  of  the  others. 
Let  us  see  how  one  ought  then  to  view  this  question. 

Let  us  give  to  each  voter  an  urn  which  contains  an 
infinite  number  of  balls,  and  let  us  suppose  that  he  dis- 
tributes them  upon  the  divers  propositions  according 
to  the  respective  probabilities  which  he  attributes  to 
them.  It  is  clear  that  the  total  number  of  balls 
expressing  certainty,  and  the  voter  being  by  the 
hypothesis  assured  that  one  of  the  propositions  ought 


130      A  PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 

to  be  true,  he  will  distribute  this  number  at  length  upon 
the  propositions.  The  problem  is  reduced  then  to  this, 
namely,  to  determine  the  combinations  in  which  the 
balls  will  be  distributed  in  such  a  manner  that  there 
may  be  more  of  them  upon  the  first  proposition  of  the 
ticket  than  upon  the  second,  more  upon  the  second 
than  upon  the  third,  etc.  ;  to  make  the  sums  of  all  the 
numbers  of  balls  relative  to  each  proposition  in  the 
divers  combinations,  and  to  divide  this  sum  by  the 
number  of  combinations;  the  quotients  will  be  the 
numbers  of  balls  that  one  ought  to  attribute  to  the 
propositions  upon  a  certain  ticket.  One  finds  by 
analysis  that  in  going  from  the  last  proposition  these 
quotients  are  among  themselves  as  the  following  quanti- 
ties :  first,  unity  divided  by  the  number  of  propositions ; 
second,  the  preceding  quantity,  augmented  by  unity, 
divided  by  the  number  of  propositions  less  one ;  third, 
this  second  quantity,  augmented  by  unity,  divided  by 
the  number  of  propositions  less  two,  and  so  on  for  the 
others.  One  will  write  then  upon  each  ticket  these 
quantities  at  the  side  of  the  corresponding  propositions, 
and  adding  the  relative  quantities  to  each  proposition 
upon  the  divers  tickets  the  sums  will  indicate  by  their 
magnitude  the  order  of  preference  which  the  assembly 
gives  to  these  propositions. 

Let  us  speak  a  word  about  the  manner  of  renewing 
assemblies  which  should  change  in  totality  in  a  definite 
number  of  years.  Ought  the  renewal  to  be  made  at 
one  time,  or  is  it  advantageous  to  divide  it  among  these 
years  ?  According  to  the  last  method  the  assembly 
would  be  formed  under  the  influence  of  the  divers 
opinions  dominant  during  the  time  of  its  renewal;  the 


SELECTIONS  AND  DECISIONS  OF  ASSEMBLIES.      I31 

opinion  which  obtained  then  would  be  probably  the 
mean  of  all  these  opinions.  The  assembly  would 
receive  thus  at  the  time  the  same  advantage  that  is 
given  to  it  by  the  extension  of  the  elections  of  its 
members  to  all  parts  of  the  territory  which  it  represents. 
Now  if  one  considers  what  experience  has  only  too 
clearly  taught,  namely,  that  elections  are  always 
directed  in  the  greatest  degree  by  dominant  opinions, 
one  will  feel  how  useful  it  is  to  temper  these  opinions, 
the  ones  by  the  others,  by  means  of  a  partial  renewal. 


CHAPTER   XIII. 

CONCERNING  THE  PROBABILITY  OF  THE  JUDG- 
MENTS OF   TRIBUNALS. 

ANALYSIS  confirms  what  simple  common  sense 
teaches  us,  namely,  the  correctness  of  judgments  is  as 
much  more  probable  as  the  judges  are  more  numerous 
and  more  enlightened.  It  is  important  then  that 
tribunals  of  appeal  should  fulfil  these  two  conditions. 
The  tribunals  of  the  first  instance  standing  in  closer 
relation  to  those  amenable  offer  to  the  higher  tribunal 
the  advantage  of  a  first  judgment  already  probable,  and 
with  which  the  latter  often  agree,  be  it  in  compromising 
or  in  desisting  from  their  claims.  But  if  the  uncertainty 
of  the  matter  in  litigation  and  its  importance  determine 
a  litigant  to  have  recourse  to  the  tribunal  of  appeals,  he 
ought  to  find  in  a  greater  probability  of  obtaining  an 
equitable  judgment  greater  security  for  his  fortune  and 
the  compensation  for  the  trouble  and  expense  which  a 
new  procedure  entails.  It  is  this  which  had  no  place 
in  the  institution  of  the  reciprocal  appeal  of  the 
tribunals  of  the  district,  an  institution  thereby  very 
prejudicial  to  the  interest  of  the  citizens.  It  would  be 
perhaps  proper  and  conformable  to  the  calculus  of 

132 


PROBABILITY  OF  THE  JUDGMENTS   OF  TRIBUNALS.   133 

probabi litres  to  demand  a  majority  of  at  least  two  votes 
in  a  tribunal  of  appeal  in  order  to  invalidate  the  sen- 
tence of  the  lower  tribunal.  One  would  obtain  this 
result  if  the  tribunal  of  appeal  being  composed  of  an 
even  number  of  judges  the  sentence  should  stand  in 
the  case  of  the  equality  of  votes. 

I  shall  consider  particularly  the  judgments  in  crimi- 
nal matters. 

In  order  to  condemn  an  accused  it  is  necessary 
without  doubt  that  the  judges  should  have  the  strongest 
proofs  of  his  offence.  But  a  moral  proof  is  never  more 
than  a  probability;  and  experience  has  only  too  clearly 
shown  the  errors  of  which  criminal  judgments,  even 
those  which  appear  to  be  the  most  just,  are  still  sus- 
ceptible. The  impossibility  of  amending  these  errors 
is  the  strongest  argument  of  the  philosophers  who  have 
wished  to  proscribe  the  penalty  of  death.  We  should 
then  be  obliged  to  abstain  from  judging  if  it  were 
necessary  for  us  to  await  mathematical  evidence.  But 
the  judgment  is  required  by  the  danger  which  would 
result  from  the  impunity  of  the  crime.  This  judgment 
reduces  itself,  if  I  am  not  mistaken,  to  the  solution  of 
the  following  question :  Has  the  proof  of  the  offence 
of  the  accused  the  high  degree  of  probability  necessary 
so  that  the  citizens  would  have  less  reason  to  doubt 
the  errors  of  the  tribunals,  if  he  is  innocent  and  con- 
demned, than  they  would  have  to  fear  his  new  crimes 
and  those  of  the  unfortunate  ones  who  would  be 
emboldened  by  the  example  of  his  impunity  if  he  were 
guilty  and  acquitted  ?  The  solution  of  this  question 
depends  upon  several  elements  very  difficult  to  ascer- 
tain. Such  is  the  eminence  of  danger  which  would 


134      A  PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 

threaten  society  if  the  criminal  accused  should  remain 
unpunished.  Sometimes  this  danger  is  so  great  that 
the  magistrate  sees  himself  constrained  to  waive  forms 
wisely  established  for  the  protection  of  innocence.  But 
that  which  renders  almost  always  this  question  insolu- 
ble is  the  impossibility  of  appreciating  exactly  the 
probability  of  the  offence  and  of  fixing  that  which  is 
necessary  for  the  condemnation  of  the  accused.  Each 
judge  in  this  respect  is  forced  to  rely  upon  his  own 
judgment.  He  forms  his  opinion  by  comparing  the 
divers  testimonies  and  the  circumstances  by  which  the 
offence  is  accompanied,  to  the  results  of  his  reflections 
and  his  experiences,  and  in  this  respect  a  long  habitude 
of  interrogating  and  judging  accused  persons  gives 
great  advantage  in  ascertaining  the  truth  in  the  midst 
of  indices  often  contradictory. 

The  preceding  question  depends  again  upon  the  care 
taken  in  the  investigation  of  the  offence;  for  one 
demands  naturally  much  stronger  proofs  for  imposing 
the  death  penalty  than  for  inflicting  a  detention  of  some 
months.  It  is  a  reason  for  proportioning  the  care  to 
the  offence,  great  care  taken  with  an  unimportant  case 
inevitably  clearing  many  guilty  ones.  A  law  which 
gives  to  the  judges  power  of  moderating  the  care  in  the 
case  of  attenuating  circumstances  is  then  conformable 
at  the  same  time  to  principles  of  humanity  towards  the 
culprit,  and  to  the  interest  of  society.  The  product  of 
the  probability  of  the  offence  by  its  gravity  being  the 
measure  of  the  danger  to  which  the  acquittal  of  the 
accused  can  expose  society,  one  would  think  that  the 
care  taken  ought  to  depend  upon  this  probability. 
This  is  done  indirectly  in  the  tribunals  where  one 


PROBABILITY  OF  THE  JUDGMENTS  OF  TRIBUNALS.   135 

retains  for  some  time  the  accused  against  whom  there 
are  very  strong  proofs,  but  insufficient  to  condemn 
him ;  in  the  hope  of  acquiring  new  light  one  does  not 
place  him  immediately  in  the  midst  of  his  fellow  citizens, 
who  would  not  see  him  again  without  great  alarm. 
But  the  arbitrariness  of  this  measure  and  the  abuse 
which  one  can  make  of  it  have  caused  its  rejection  in 
the  countries  where  one  attaches  the  greatest  price  to 
individual  liberty. 

Now  what  is  the  probability  that  the  decision  of  a 
tribunal  which  can  condemn  only  by  a  given  majority 
will  be  just,  that  is  to  say,  conform  to  the  true  solution 
of  the  question  proposed  above  ?  This  important 
problem  well  solved  will  give  the  means  of  compar- 
ing among  themselves  the  different  tribunals.  The 
majority  of  a  single  vote  in  a  numerous  tribunal  indi- 
cates that  the  affair  in  question  is  very  doubtful ;  the 
condemnation  of  the  accused  would  be  then  contrary 
to  the  principles  of  humanity,  protectors  of  innocence. 
The  unanimity  of  the  judges  would  give  very  strong 
probability  of  a  just  decision ;  but  in  abstaining  from  it 
too  many  guilty  ones  would  be  acquitted.  It  is  neces- 
sary, then,  either  to  limit  the  number  of  judges,  if  one 
wishes  that  they  should  be  unanimous,  or  increase  the 
majority  necessary  for  a  condemnation,  when  the  tri- 
bunal becomes  more  numerous.  I  shall  attempt  to 
apply  calculus  to  this  subject,  being  persuaded  that  it 
is  always  the  best  guide  when  one  bases  it  upon  the 
data  which  common  sense  suggests  to  us. 

The  probability  that  the  opinion  of  each  judge  is  just 
enters  as  the  principal  element  into  this  calculation. 
If  in  a  tribunal  of  a  thousand  and  one  judges,  five 


136     A  PHILOSOPHICAL  ESSAY  OV  PROBABILITIES. 

hundred  and  one  are  of  one  opinion,  and  five  hundred 
are  of  the  contrary  opinion,  it  is  apparent  that  the 
probability  of  the  opinion  of  each  judge  surpasses  very 
little  £;  for  supposing  it  obviously  very  large  a  single 
vote  of  difference  would  be  an  improbable  event.  But 
if  the  judges  are  unanimous,  this  indicates  in  the  proofs 
that  degree  of  strength  which  entails  conviction;  the 
probability  of  the  opinion  of  each  judge  is  then  very 
near  unity  or  certainty,  provided  that  the  passions  or 
the  ordinary  prejudices  do  not  affect  at  the  same  time 
all  the  judges.  Outside  of  these  cases  the  ratio  of  the 
votes  for  or  against  the  accused  ought  alone  to  deter- 
mine this  probability.  I  suppose  thus  that  it  can  vary 
from  £  to  unity,  but  that  it  cannot  be  below  £.  If  that 
were  not  the  case  the  decision  of  the  tribunal  would  be 
as  insignificant  as  chance ;  it  has  value  only  in  so  far 
as  the  opinion  of  the  judge  has  a  greater  tendency  to 
truth  than  to  error.  It  is  thus  by  the  ratio  of  the 
numbers  of  votes  favorable,  and  contrary  to  the  accused, 
that  I  determine  the  probability  of  this  opinion. 

These  data  suffice  to  ascertain  the  general  expression 
of  the  probability  that  the  decision  of  a  tribunal  judging 
by  a  known  majority  is  just.  In  the  tribunals  where 
of  eight  judges  five  votes  would  be  necessary  for  the 
condemnation  of  an  accused,  the  probability  of  the 
error  to  be  feared  in  the  justice  of  the  decision  would 
surpass  ^.  If  the  tribunal  should  be  reduced  to  six 
members  who  are  able  to  condemn  only  by  a  plurality 
of  four  votes,  the  probability  of  the  error  to  be  feared 
would  be  below  ^.  There  would  be  then  for  the 
accused  an  advantage  in  this  reduction  of  the  tribunal. 
In  both  cases  the  majority  required  is  the  same  and  is 


PROBABILITY  OF   THE  JUDGMENTS   OF  TRIBUNALS.    13? 

equal  to  two.  Thus  the  majority  remaining  constant, 
the  probability  of  error  increases  with  the  number  of 
judges ;  this  is  general  whatever  may  be  the  majority 
required,  provided  that  it  remains  the  same.  Taking, 
then,  for  the  rule  the  arithmetical,  ratio,  the  accused 
finds  himself  in  a  position  less  and  less  advantageous 
in  the  measure  that  the  tribunal  becomes  more  numer- 
ous. One  might  believe  that  in  a  tribunal  where  one 
might  demand  a  majority  of  twelve  votes,  whatever 
the  number  of  the  judges  was,  the  votes  of  the  minority, 
neutralizing  an  equal  number  of  votes  of  the  majority, 
the  twelve  remaining  votes  would  represent  the 
unanimity  of  a  jury  of  twelve  members,  required  in 
England  for  the  condemnation  of  an  accused ;  but  one 
would  be  greatly  mistaken.  Common  sense  shows 
that  there  is  a  difference  between  the  decision  of  a 
tribunal  of  two  hundred  and  twelve  judges,  of  which 
one  hundred  and  twelve  condemn  the  accused,  while 
one  hundred  acquit  him,  and  that  of  a  tribunal  of 
twelve  judges  unanimous  for  condemnation.  In  the 
first  case  the  hundred  votes  favorable  to  the  accused 
warrant  in  thinking  that  the  proofs  are  far  from  attain- 
ing the  degree  of  strength  which  entails  conviction ;  in 
the  second  case,  the  unanimity  of  the  judges  leads  to 
the  belief  that  they  have  attained  this  degree.  But 
simple  common  sense  does  not  suffice  at  all  to  appre- 
ciate the  extreme  difference  of  the  probability  of  error 
in  the  two  cases.  It  is  necessary  then  to  recur  to 
calculus,  and  one  finds  nearly  one  fifth  for  the  prob- 
ability of  error  in  the  first  case,  and  only  -^-^  for  this 
probability  in  the  second  case,  a  probability  which  is 
not  one  thousandth  of  the  first.  It  is  a  confirmation 


I38      A  PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 

of  the  principle  that  the  arithmetical  ratio  is  unfavorable 
to  the  accused  when  the  number  of  judges  increases. 
On  the  contrary,  if  one  takes  for  a  rule  the  geometrical 
ratio,  the  probability  of  the  error  of  the  decision 
diminishes  when  the  number  of  judges  increases.  For 
example,  in  the  tribunals  which  can  condemn  only  by 
a  plurality  of  two  thirds  of  the  votes,  the  probability  of 
the  error  to  be  feared  is  nearly  one  fourth  if  the 
number  of  the  judges  is  six;  it  is  below  %  if  this  number 
is  increased  to  twelve.  Thus  one  ought  to  be  governed 
neither  by  the  arithmetical  ratio  nor  by  the  geometrical 
ratio  if  one  wishes  that  the  probability  of  error  should 
never  be  above  nor  below  a  given  fraction. 

But  what  fraction  ought  to  be  determined  upon  ?  It 
is  here  that  the  arbitrariness  begins  and  the  tribunals 
offer  in  this  regard  the  greatest  variety.  In  the  special 
tribunals  where  five  of  the  eight  votes  suffice  for  the 
condemnation  of  the  accused,  the  probability  of  the 
error  to  be  feared  in  regard  to  justice  of  the  judgment 
is  ^6/ff,  or  more  than  ^.  The  magnitude  of  this  fraction 
is  dreadful ;  but  that  which  ought  to  reassure  us  a  little 
is  the  consideration  that  most  frequently  the  judge  who 
acquits  an  accused  does  not  regard  him  as  innocent; 
he  pronounces  solely  that  it  is  not  attained  by  proofs 
sufficient  for  condemnation.  One  is  especially  reassured 
by  the  pity  which  nature  has  placed  in  the  heart  of  man 
and  which  disposes  the  mind  to  see  only  with  reluc- 
tance a  culprit  in  the  accused  submitted  to  his  judg- 
ment. This  sentiment,  more  active  in  those  who  have 
not  the  habitude  of  criminal  judgments,  compensates 
for  the  inconveniences  attached  to  the  inexperience  of 
the  jurors.  In  a  jury  of  twelve  members,  if  the  plurality 


PROBABILITY  OF  THE  JUDGMENTS   OF   TRIBUNALS.   139 

demanded  for  the  condemnation  is  eight  of  twelve 
votes,  the  probability  of  the  error  to  be  feared  |fff ,  or 
a  little  more  than  one  eighth,  it  is  almost  -fa  if  this 
plurality  consists  of  nine  votes.  In  the  case  of  una- 
nimity the  probability  of  the  error  to  be  feared  is  -g-T17j, 
that  is  to  say,  more  than  a  thousand  times  less  than 
in  our  juries.  This  supposes  that  the  unanimity  results 
only  from  proofs  favorable  or  contrary  to  the  accused ; 
but  motives  that  are  entirely  strange,  ought  oftentimes 
to  concur  in  producing  it,  when  it  is  imposed  upon  the 
jury  as  a  necessary  condition  of  its  judgment.  Then 
its  decisions  depending  upon  the  temperament,  the 
character,  the  habits  of  the  jurors,  and  the  circum- 
stances in  which  they  are  placed,  they  are  sometimes 
contrary  to  the  decisions  which  the  majority  of  the  jury 
would  have  made  if  they  had  listened  only  to  the 
proofs ;  this  seems  to  me  to  be  a  great  fault  of  this 
manner  of  judging. 

The  probability  of  the  decision  is  too  feeble  in  our 
juries,  and  I  think  that  in  order  to  give  a  sufficient 
guarantee  to  innocence,  one  ought  to  demand  at  least 
a  plurality  of  nine  votes  in  twelve. 


CHAPTER    XIV. 

CONCERNING  TABLES  OF  MORTALITY,  AND  OF 
MEAN  DURATIONS  OF  LIFE,  OF  MARRIAGES, 
AND  OF  ASSOCIATIONS. 

THE  manner  of  preparing  tables  of  mortality  is  very 
simple.  One  takes  in  the  civil  registers  a  great  num- 
ber of  individuals  whose  birth  and  death  are  indicated. 
One  determines  how  many  of  these  individuals  have 
died  in  the  first  year  of  their  age,  how  many  in  the 
second  year,  and  so  on.  It  is  concluded  from  these 
the  number  of  individuals  living  at  the  commencement 
of  each  year,  and  this  number  is  written  in  the  table  at 
the  side  of  that  which  indicates  the  year.  Thus  one 
writes  at  the  side  of  zero  the  number  of  births ;  at  the 
side  of  the  year  I  the  number  of  infants  who  have 
attained  one  year;  at  the  side  of  the  year  2  the  number 
of  infants  who  have  attained  two  years,  and  so  on  for 
the  rest.  But  since  in  the  first  two  years  of  life  the 
mortality  is  very  great,  it  is  necessary  for  the  sake  of 
greater  exactitude  to  indicate  in  this  first  age  the 
number  of  survivors  at  the  end  of  each  half  year. 

If  we  divide  the  sum  of  the  years  of  the  life  of  all 
the  individuals  inscribed  in  a  table  of  mortality  by  the 

140 


CONCERNING    TABLES  OF  MORTALITY,  ETC.        U* 

number  of  these  individuals  we  shall  have  the  mean 
duration  of  life  which  corresponds  to  this  table.  For 
this,  we  will  multiply  by  a  half  year  the  number  of 
deaths  in  the  first  year,  a  number  equal  to  the  differ- 
ence of  the  numbers  of  individuals  inscribed  at  the  side 
of  the  years  o  and  I.  Their  mortality  being  distributed 
over  the  entire  year  the  mean  duration  of  their  life  is 
only  a  half  year.  We  will  multiply  by  a  year  and  a 
half  the  number  of  deaths  in  the  second  year;  by  two 
years  and  a  half  the  number  of  deaths  in  the  third  year ; 
and  so  on.  The  sum  of  these  products  divided  by  the 
number  of  births  will  be  the  mean  duration  of  life.  It 
is  easy  to  conclude  from  this  that  we  will  obtain  this 
duration,  by  making  the  sum  of  the  numbers  inscribed 
in  the  table  at  the  side  of  each  year,  dividing  it  by  the 
number  of  births  and  subtracting  one  half  from  the 
quotient,  the  year  being  taken  as  unity.  The  mean 
duration  of  life  that  remains,  starting  from  any  age,  is 
determined  in  the  same  manner,  working  upon  the 
number  of  individuals  who  have  arrived  at  this  age,  as 
has  just  been  done  with  the  number  of  births.  But  it 
is  not  at  the  moment  of  birth  that  the  mean  duration 
of  life  is  the  greatest;  it  is  when  one  has  escaped  the 
dangers  of  infancy  and  it  is  then  about  forty-three 
years.  The  probability  of  arriving  at  a  certain  age, 
starting  from  a  given  age  is  equal  to  the  ratio  of  the 
two  numbers  of  individuals  indicated  in  the  table  at 
these  two  ages. 

The  precision  of  these  results  demands  that  for  the 
formation  of  tables  we  should  employ  a  very  great 
number  of  births.  Analysis  gives  then  very  simple 
formulae  for  appreciating  the  probability  that  the  num- 


142      A  PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 

bers  indicated  in  these  tables  will  vary  from  the  truth 
only  within  narrow  limits.  We  see  by  these  formulae 
that  the  interval  of  the  limits  diminishes  and  that  the 
probability  increases  in  proportion  as  we  take  into  con- 
sideration more  births;  so  that  the  tables  would  repre- 
sent exactly  the  true  law  of  mortality  if  the  number  of 
births  employed  were  infinite. 

A  table  of  mortality  is  then  a  table  of  the  probability 
of  human  life.  The  ratio  of  the  individuals  inscribed 
at  the  side  of  each  year  to  the  number  of  births  is  the 
probability  that  a  new  birth  will  attain  this  year.  As 
we  estimate  the  value  of  hope  by  making  a  sum  of  the 
products  of  each  benefit  hoped  for,  by  the  probability 
of  obtaining  it,  so  we  can  equally  evaluate  the  mean 
duration  of  life  by  adding  the  products  of  each  year 
by  half  the  sum  of  the  probabilities  of  attaining  the 
commencement  and  the  end  of  it,  which  leads  to  the 
result  found  above.  But  this  manner  of  viewing  the 
mean  duration  of  life  has  the  advantage  of  showing 
that  in  a  stationary  population,  that  is  to  say,  such  that 
the  number  of  births  equals  that  of  deaths,  the  mean 
duration  of  life  is  the  ratio  itself  of  the  population  to 
the  annual  births;  for  the  population  being  supposed 
stationary,  the  number  of  individuals  of  an  age  com- 
prised between  two  consecutive  years  of  the  table  is 
equal  to  the  number  of  annual  births,  multiplied  by 
half  the  sum  of  the  probabilities  of  attaining  these 
years ;  the  sum  of  all  these  products  will  be  then  the 
entire  population.  Now  it  is  easy  to  see  that  this  sum, 
divided  by  the  number  of  annual  births,  coincides  with 
the  mean  duration  of  life  as  we  have  just  defined  it. 

It  is  easy  by  means  of  a  table  of  mortality  to  form 


CONCERNING    TABLES  OF  MORTALITY,  ETC.        143 

the  corresponding  table  of  the  population  supposed  to 
be  stationary.  For  this  we  take  the  arithmetical  means 
of  the  numbers  of  the  table  of  mortality  corresponding 
to  the  ages  zero  and  one  year,  one  and  two  years,  two 
and  three  years,  etc.  The  sum  of  all  these  means  is 
the  entire  population;  it  is  written  at  the  side  of  the 
age  zero.  There  is  subtracted  from  this  sum  the  first 
mean  and  the  remainder  is  the  number  of  individuals 
of  one  year  and  upwards;  it  is  written  at  the  side  of 
the  year  I .  There  is  subtracted  from  this  first  re- 
mainder the  second  mean ;  this  second  remainder  is 
the  number  of  individuals  of  two  years  and  upwards ; 
it  is  written  at  the  side  of  the  year  2,  and  so  on. 

So  many  variable  causes  influence  mortality  that  the 
tables  which  represent  it  ought  to  be  changed  accord- 
ing to  place  and  time.  The  divers  states  of  life  offer 
in  this  regard  appreciable  differences  relative  to  the 
fatigues  and  the  dangers  inseparable  from  each  state 
and  of  which  it  is  indispensable  to  keep  account  in  the 
calculations  founded  upon  the  duration  of  life.  But 
these  differences  have  not  been  sufficiently  observed. 
Some  day  they  will  be  and  then  will  be  known  what 
sacrifice  of  life  each  profession  demands  and  one  will 
profit  by  this  knowledge  to  diminish  the  dangers. 

The  greater  or  less  salubrity  of  the  sun,  its  elevation, 
its  temperature,  the  customs  of  the  inhabitants,  and  the 
operations  of  governments  have  a  considerable  influence 
upon  mortality.  But  it  is  always  necessary  to  precede 
the  investigation  of  the  cause  of  the  differences  observed 
by  that  of  the  probability  with  which  this  cause  is  indi- 
cated. Thus  the  ratio  of  the  population  to  annual 
births,  which  one  has  seen  raised  in  France  to  twenty- 


144      A  PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

eight  and  one  third,  is  not  equal  to  twenty-five  in  the 
ancient  duchy  of  Milan.  These  ratios,  both  established 
upon  a  great  number  of  births,  do  not  permit  of  calling 
into  question  the  existence  among  the  Milanese  of  a 
special  cause  of  mortality,  which  it  is  of  moment  for 
the  government  of  our  country  to  investigate  and 
remove. 

The  ratio  of  the  population  to  the  births  would 
increase  again  if  we  could  diminish  and  remove  certain 
dangerous  and  widely  spread  maladies.  This  has 
happily  been  done  for  the  smallpox,  at  first  by  the 
inoculation  of  this  disease,  then  in  a  manner  much 
more  advantageous,  by  the  inoculation  of  vaccine,  the 
inestimable  discovery  of  Jenner,  who  has  thereby 
become  one  of  the  greatest  benefactors  of  humanity. 

The  smallpox  has  this  in  particular,  namely,  that 
the  same  individual  is  not  twice  affected  by  it,  or  at 
least  such  cases  are  so  rare  that  they  may  be  abstracted 
from  the  calculation.  This  malady,  from  which  few 
escaped  before  the  discovery  of  vaccine,  is  often  fatal 
and  causes  the  death  of  one  seventh  of  those  whom  it 
attacks.  Sometimes  it  is  mild,  and  experience  has 
taught  that  it  can  be  given  this  latter  character  by 
inoculating  it  upon  healthy  persons,  prepared  for  it 
by  a  proper  diet  and  in  a  favorable  season.  Then  the 
ratio  of  the  individuals  who  die  to  the  inoculated 
ones  is  not  one  three  hundredth.  This  great  advan- 
tage of  inoculation,  joined  to  those  of  not  altering  the 
appearance  and  of  preserving  from  the  grievous  conse- 
quences which  the  natural  smallpox  often  brings, 
caused  it  to  be  adopted  by  a  great  number  of  persons. 
The  practice  was  strongly  recommended,  but  it  was 


CONCERNING    TABLES  OF  MORTALITY,  ETC.        145 

strongly  combated,  as  is  nearly  always  the  case  in 
things  subject  to  inconvenience.  In  the  midst  of  this 
dispute  Daniel  Bernoulli  proposed  to  submit  to  the 
calculus  of  probabilities  the  influence  of  inoculation 
upon  the  mean  duration  of  life.  Since  precise  data  of 
the  mortality  produced  by  the  smallpox  at  the  various 
ages  of  life  were  lacking,  he  supposed  that  the  danger 
of  having  this  malady  and  that  of  dying  of  it  are  the 
same  at  every  age.  By  means  of  these  suppositions  he 
succeeded  by  a  delicate  analysis  in  converting  an 
ordinary  table  of  mortality  into  that  which  would  be 
used  if  smallpox  did  not  exist,  or  if  it  caused  the 
death  of  only  a  very  small  number  of  those  affected,  and 
he  concludes  from  it  that  inoculation  would  augment 
by  three  years  at  least  the  mean  duration  of  life,  which 
appeared  to  him  beyond  doubt  the  advantage  of  this 
operation.  D'Alembert  attacked  the  analysis  of  Ber- 
noulli: at  first  in  regard  to  the  uncertainty  of  his 
two  hypotheses,  then  in  regard  to  its  insufficiency  in 
this,  that  no  comparison  was  made  of  the  immediate 
danger,  although  very  small,  of  dying  of  inoculation,  to 
the  very  great  but  very  remote  danger  of  succumbing 
to  natural  smallpox.  This  consideration,  which  dis- 
appears when  one  considers  a  great  number  of  indi- 
viduals, is  for  this  reason  immaterial  for  governments 
and  the  advantages  of  inoculation  for  them  still  remain  ; 
but  it  is  of  great  weight  for  the  father  of  a  family  who 
must  fear,  in  having  his  children  inoculated,  to  see  that 
one  perish  whom  he  holds  most  dear  and  to  be  the 
cause  of  it.  Many  parents  were  restrained  by  this  fear, 
which  the  discovery  of  vaccine  has  happily  dissipated. 
By  one  of  those  mysteries  which  nature  offers  to  us  so 


146      A  PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 

frequently,  vaccine  is  a  preventive  of  smallpox  just  as 
certain  as  variolar  virus,  and  there  is  no  danger  at  all ; 
it  does  not  expose  to  any  malady  and  demands  only 
very  little  care.  Therefore  the  practice  of  it  has  spread 
quickly;  and  to  render  it  universal  it  remains  only  to 
overcome  the  natural  inertia  of  the  people,  against 
which  it  is  necessary  to  strive  continually,  even  when 
it  is  a  question  of  their  dearest  interests. 

The  simplest  means  of  calculating  the  advantage 
which  the  extinction  of  a  malady  would  produce  con- 
sists in  determining  by  observation  the  number  of  indi- 
viduals of  a  given  age  who  die  of  it  each  year  and 
subtracting  this  number  from  the  number  of  deaths  at 
the  same  age.  The  ratio  of  the  difference  to  the  total 
number  of  individuals  of  the  given  age  would  be  the 
probability  of  dying  in  the  year  at  this  age  if  the 
malady  did  not  exist.  Making,  then,  a  sum  of  these 
probabilities  from  birth  up  to  any  given  age,  and  sub- 
tracting this  sum  from  unity,  the  remainder  will  be  the 
probability  of  living  to  that  age  corresponding  to  the 
extinction  of  the  malady.  The  series  of  these  prob- 
abilities will  be  the  table  of  mortality  relative  to  this 
hypothesis,  and  we  may  conclude  from  it,  by  what 
precedes,  the  mean  duration  of  life.  It  is  thus  that 
Duvilard  has  found  that  the  increase  of  the  mean  dura- 
tion of  life,  due  to  inoculation  with  vaccine,  is  three 
years  at  the  least.  An  increase  so  considerable  would 
produce  a  very  great  increase  in  the  population  if  the 
latter,  for  other  reasons,  were  not  restrained  by  the 
relative  diminution  of  subsistences. 

It  is  principally  by  the  lack  of  subsistences  that  the 
progressive  march  of  the  population  is  arrested.  In 


CONCERNING    TABLES   OF  MORTALITY,  ETC.        14? 

all  kinds  of  animals  and  vegetables,  nature*  tends  with- 
out ceasing  to  augment  the  number  of  individuals  until 
they  are  on  a  level  of  the  means  of  subsistence.  In 
the  human  race  moral  causes  have  a  great  influence 
upon  the  population.  If  easy  clearings  of  the  forest 
can  furnish  an  abundant  nourishment  for  new  genera- 
tions, the  certainty  of  being  able  to  support  a  numerous 
family  encourages  marriages  and  renders  them  more 
productive.  Upon  the  same  soil  the  population  and 
the  births  ought  to  increase  at  the  same  time  simul- 
taneously in  geometric  progression.  But  when  clear- 
ings become  more  difficult  and  more  rare  then  the 
increase  of  population  diminishes;  it  approaches  con- 
tinually the  variable  state  of  subsistences,  making 
oscillations  about  it  just  as  a  pendulum  whose  periodicity 
is  retarded  by  changing  the  point  of  suspension,  oscil- 
lates about  this  point  by  virtue  of  its  own  weight.  It 
is  difficult  to  evaluate  the  maximum  increase  of  the 
population ;  it  appears  after  observations  that  in  favor- 
able circumstances  the  population  of  the  human  race 
would  be  doubled  every  fifteen  years.  We  estimate 
that  in  North  America  the  period  of  this  doubling  is 
twenty-two  years.  In  this  state  of  things,  the  popula- 
tion, births,  marriages,  mortality,  all  increase  accord- 
ing to  the  same  geometric  progression  of  which  we  have 
the  constant  ratio  of  consecutive  terms  by  the  observa- 
tion of  annual  births  at  two  epochs. 

By  means  of  a  table  of  mortality  representing  the 
probabilities  of  human  life,  we  may  determine  the 
duration  of  marriages.  Supposing  in  order  to  simplify 
the  matter  that  the  mortality  is  the  same  for  the  two 
sexes,  we  shall  obtain  the  probability  that  the  marriage 


U$      A  PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 

will  subsist  one  year,  or  two,  or  three,  etc.,  by  forming 
a  series  of  fractions  whose  common  denominator  is  the 
product  of  the  two  numbers  of  the  table  corresponding 
to  the  ages  of  the  consorts,  and  whose  numerators  are 
the  successive  products  of  the  numbers  corresponding 
to  these  ages  augmented  by  one,  by  two,  by  three, 
etc.,  years.  The  sum  of  these  fractions  augmented  by 
one  half  will  be  the  mean  duration  of  marriage,  the 
year  being  taken  as  unity.  It  is  easy  to  extend  the 
same  rule  to  the  mean  duration  of  an  association  formed 
of  three  or  of  a  greater  number  of  individuals. 


CHAPTER    XV. 

CONCERNING  THE  BENEFITS  OF  INSTITUTIONS 
WHICH  DEPEND  UPON  THE  PROBABILITY  OF 
EVENTS. 

LET  us  recall  here  what  has  been  said  in  speaking 
of  hope.  It  has  been  seen  that  in  order  to  obtain  the 
advantage  which  results  from  several  simple  events,  of 
which  the  ones  produce  a  benefit  and  the  others  a  loss, 
it  is  necessary  to  add  the  products  of  the  probability  of 
each  favorable  event  by  the  benefit  which  it  procures, 
and  subtract  from  their  sum  that  of  the  products  of  the 
probability  of  each  unfavorable  event  by  the  loss  which 
is  attached  to  it.  But  whatever  may  be  the  advantage 
expressed  by  the  difference  of  these  sums,  a  single 
event  composed  of  these  simple  events  does  not 
guarantee  against  the  fear  of  experiencing  a  loss. 
One  imagines  that  this  fear  ought  to  decrease  when 
one  multiplies  the  compound  event.  The  analysis  of 
probabilities  leads  to  this  general  theorem. 

By  the  repetition  of  an  advantageous  event,  simple 
or  compound,  the  real  benefit  becomes  more  and  more 
probable  and  increases  without  ceasing;  it  becomes 
certain  in  the  hypothesis  of  an  infinite  number  of  repe- 

149 


150      A  PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

titions;  and  dividing  it  by  this  number  the  quotient  or 
the  mean  benefit  of  each  event  is  the  mathematical 
hope  itself  or  the  advantage  relative  to  the  event.  It 
is  the  same  with  a  loss  which  becomes  certain  in  the 
long  run,  however  small  the  disadvantage  of  the  event 
may  be. 

This  theorem  upon  benefits  and  losses  is  analogous 
to  those  which  we  have  already  given  upon  the  ratios 
which  are  indicated  by  the  indefinite  repetition  of 
events  simple  or  compound;  and,  like  them,  it  proves 
that  regularity  ends  by  establishing  itself  even  in  the 
things  which  are  most  subordinated  to  that  which  we 
name  hazard. 

When  the  events  are  in  great  number,  analysis  gives 
another  very  simple  expression  of  the  probability  that 
the  benefit  will  be  comprised  within  determined  limits. 
This  is  the  expression  which  enters  again  into  the 
general  law  of  probability  given  above  in  speaking 
of  the  probabilities  which  result  from  the  indefinite 
multiplication  of  events. 

The  stability  of  institutions  which  are  based  upon 
probabilities  depends  upon  the  truth  of  the  preceding 
theorem.  But  in  order  that  it  may  be  applied  to  them 
it  is  necessary  that  those  institutions  should  multiply 
these  advantageous  events  for  the  sake  of  numerous 
things. 

There  have  been  based  upon  the  probabilities  of 
human  life  divers  institutions,  such  as  life  annuities  and 
tontines.  The  most  general  and  the  most  simple 
method  of  calculating  the  benefits  and  the  expenses  of 
these  institutions-  consists  in  reducing  these  to  actual 
amounts.  The  annual  interest  of  unity  is  that  which 


INSTITUTIONS  BASED  UPON  PROBABILITIES.       151 

is  called  the  rate  of  interest.  At  the  end  of  each  year 
an  amount  acquires  for  a  factor  unity  plus  the  rate  of 
interest;  it  increases  then  according  to  a  geometrical 
progression  of  which  this  factor  is  the  ratio.  Thus  in 
the  course  of  time  it  becomes  immense.  If,  for  exam- 
ple, the  rate  of  interest  is  -£-$  or  five  per  cent,  the  capital 
doubles  very  nearly  in  fourteen  years,  quadruples  in 
twenty-nine  years,  and  in  less  than  three  centuries  it 
becomes  two  million  times  larger. 

An  increase  so  prodigious  has  given  birth  to  the  idea 
of  making  use  of  it  in  order  to  pay  off  the  public  debt. 
One  forms  for  this  purpose  a  sinking  fund  to  which  is 
devoted  an  annual  fund  employed  for  the  redemption 
of  public  bills  and  without  ceasing  increased  by  the 
interest  of  the  bills  redeemed.  It  is  clear  that  in  the 
long  run  this  fund  will  absorb  a  great  part  of  the 
national  debt.  If,  when  the  needs  of  the  State  make 
a  loan  necessary,  a  part  of  this  loan  is  devoted  to  the 
increasing  of  the  annual  sinking  fund,  the  variation  of 
public  bills  will  be  less;  the  confidence  of  the  lenders 
and  the  probability  of  retiring  without  loss  of  capital 
loaned  when  one  desires  will  be  augmented  and  will 
render  the  conditions  of  the  loan  less  onerous.  Favor- 
able experiences  have  fully  confirmed  these  advantages. 
But  the  fidelity  in  engagements  and  the  stability,  so 
necessary  to  the  success  of  such  institutions,  can  be 
guaranteed  only  by  a  government  in  which  the  legisla- 
tive power  is  divided  among  several  independent 
powers.  The  confidence  which  the  necessary  coopera- 
tion of  these  powers  inspires,  doubles  the  strength  of 
the  State,  and  the  sovereign  himself  gains  then  in  legal 
power  more  than  he  loses  in  arbitrary  power. 


152      A  PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 

It  results  from  that  which  precedes  that  the  actual 
capital  equivalent  to  a  sum  which  is  to  be  paid  only 
after  a  certain  number  of  years  is  equal  to  this  sum 
multiplied  by  the  probability  that  it  will  be  paid  at  that 
time  and  divided  by  unity  augmented  by  the  rate  of 
interest  and  raised  to  a  power  expressed  by  the  number 
of  these  years. 

It  is  easy  to  apply  this  principle  to  life  annuities  upon 
one  or  several  persons,  and  to  savings  banks,  and  to 
assurance  societies  of  any  nature.  Suppose  that  one 
proposes  to  form  a  table  of  life  annuities  according  to 
a  given  table  of  mortality.  A  life  annuity  payable  at 
the  end  of  five  years,  for  example,  and  reduced  to  an 
actual  amount  is,  by  this  principle,  equal  to  the  product 
of  the  two  following  quantities,  namely,  the  annuity 
divided  by  the  fifth  power  of  unity  augmented  by  the 
rate  of  interest  and  the  probability  of  paying  it.  This 
probability  is  the  inverse  ratio  of  the  number  of  indi- 
viduals inscribed  in  the  table  opposite  to  the  age  of  that 
one  who  settles  the  annuity  to  the  number  inscribed 
opposite  to  this  age  augmented  by  five  years.  Form- 
ing, then,  a  series  of  fractions  whose  denominators  are 
the  products  of  the  number  of  persons  indicated  in  the 
table  of  mortality  as  living  at  the  age  of  that  one  who 
settles  the  annuity,  by  the  successive  powers  of  unity 
augmented  by  the  rate  of  interest,  and  whose  numera- 
tors are  the  products  of  the  annuity  by  the  number  of 
persons  living  at  the  same  age  augmented  successively 
by  one  year,  by  two  years,  etc.,  the  sum  of  these 
fractions  will  be  the  amount  required  for  the  life  annuity 
at  that  age. 

Let  us  suppose  that  a  person  wishes  by  means  of  a 


INSTITUTIONS  BASED   UPON  PROBABILITIES.       153 

life  annuity  to  assure  to  his  heirs  an  amount  payable 
at  the  end  of  the  year  of  his  death.  In  order  to  deter- 
mine the  value  of  this  annuity,  one  may  imagine  that 
the  person  borrows  in  life  at  a  bank  this  capital  and 
that  he  places  it  at  perpetual  interest  in  the  same  bank. 
It  is  clear  that  this  same  capital  will  be  due  by  the 
bank  to  his  heirs  at  the  end  of  the  year  of  his  death ; 
but  he  will  have  paid  each  year  only  the  excess  of  the 
life  interest  over  the  perpetual  interest.  The  table  of 
life  annuities  will  then  show  that  which  the  person 
ought  to  pay  annually  to  the  bank  in  order  to  assure 
this  capital  after  his  death. 

Maritime  assurance,  that  against  fire  and  storms,  and 
generally  all  the  institutions  of  this  kind,  are  computed 
on  the  same  principles.  A  merchant  having  vessels 
at  sea  wishes  to  assure  their  value  and  that  of  their 
cargoes  against  the  dangers  that  they  may  run ;  in  order 
to  do  this,  he  gives  a  sum  to  a  company  which  becomes 
responsible  to  him  for  the  estimated  value  of  his 
cargoes  and  his  vessels.  The  ratio  of  this  value  to  the 
sum  which  ought  to  be  given  for  the  price  of  the  assur- 
ance depends  upon  the  dangers  to  which  the  vessels 
are  exposed  and  can  be  appreciated  only  by  numerous 
observations  upon  the  fate  of  vessels  which  have  sailed 
from  port  for  the  same  destination. 

If  the  persons  assured  should  give  to  the  assurance 
company  only  the  sum  indicated  by  the  calculus  of 
probabilities,  this  company  would  not  be  able  to  pro- 
vide for  the  expenses  of  its  institution ;  it  is  necessary 
then  that  they  should  pay  a  sum  much  greater  than  the 
cost  of  such  insurance.  What  then  is  their  advantage  ? 
It  is  here  that  the  consideration  of  the  moral  disadvan- 


154      A  PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

tage  attached  to  an  uncertainty  becomes  necessary. 
One  conceives  that  the  fairest  game  becomes,  as  has 
already  been  seen,  disadvantageous,  because  the  player 
exchanges  a  certain  stake  for  an  uncertain  benefit; 
assurance  by  which  one  exchanges  the  uncertain  for 
the  certain  ought  to  be  advantageous.  It  is  indeed 
this  which  results  from  the  rule  which  we  have  given 
above  for  determining  moral  hope  and  by  which  one 
sees  moreover  how  far  the  sacrifice  may  extend  which 
ought  to  be  made  to  the  assurance  company  by 
reserving  always  a  moral  advantage.  This  company 
can  then  in  procuring  this  advantage  itself  make  a 
great  benefit,  if  the  number  of  the  assured  persons  is 
very  large,  a  condition  .  necessary  to  its  continued 
existence.  Then  its  benefits  become  certain  and  the 
mathematical  and  moral  hopes  coincide;  for  analysis 
leads  to  this  general  theorem,  namely,  that  if  the 
expectations  are  very  numerous  the  two  hopes  approach 
each  other  without  ceasing  and  end  by  coinciding  in 
the  case  of  an  infinite  number. 

We  have  said  in  speaking  of  mathematical  and  moral 
hopes  that  there  is  a  moral  advantage  in  distributing 
the  risks  of  a  benefit  which  one  expects  over  several  of 
its  parts.  Thus  in  order  to  send  a  sum  of  money  to  a 
distant  part  it  is  much  better  to  send  it  on  several 
vessels  than  to  expose  it  on  one.  This  one  does  by 
means  of  mutual  assurances.  If  two  persons,  each 
having  the  same  sum  upon  two  different  vessels  which 
have  sailed  from  the  same  port  to  the  same  destination, 
agree  to  divide  equally  all  the  money  which  may 
arrive,  it  is  clear  that  by  this  agreement  each  of  them 
divides  equally  between  the  two  vessels  the  sum  which 


INSTITUTIONS  BASED   UPON  PROBABILITIES.       155 

he  expects.  Indeed  this  kind  of  assurance  always 
leaves  uncertainty  as  to  the  loss  which  one  may  fear. 
But  this  uncertainty  diminishes  in  proportion  as  the 
number  of  policy-holders  increases ;  the  moral  advan- 
tage increases  more  and  more  and  ends  by  coinciding 
with  the  mathematical  advantage,  its  natural  limit. 
This  renders  the  association  of  mutual  assurances  when 
it  is  very  numerous  more  advantageous  to  the  assured 
ones  than  the  companies  of  assurance  which,  in  pro- 
portion to  the  benefit  that  they  give,  give  a  moral 
advantage  always  inferior  to  the  mathematical  advan- 
tage. But  the  surveillance  of  their  administration  can 
balance  the  advantage  of  the  mutual  assurances.  All 
these  results  are,  as  has  already  been  seen,  independent 
of  the  law  which  expresses  the  moral  advantage. 

One  may  look  upon  a  free  people  as  a  great  asso- 
ciation whose  members  secure  mutually  their  proper- 
ties by  supporting  proportionally  the  charges  of  this 
guaranty.  The  confederation  of  several  peoples  would 
give  to  them  advantages  analogous  to  those  which  each 
individual  enjoys  in  the  society.  A  congress  of  their 
representatives  would  discuss  objects  of  a  utility  com- 
mon to  all  and  without  doubt~the  system  of  weights, 
measures,  and  moneys  proposed  by  the  French  sci- 
entists would  be  adopted  in  this  congress  as  one  of 
the  things  most  useful  to  commerical  relations. 

Among  the  institutions  founded  upon  the  probabilities 
of  human  life  the  better  ones  are  those  in  which,  by 
means  of  a  light  sacrifice  of  his  revenue,  one  assures 
his  existence  and  that  of  his  family  for  a  time  when 
one  ought  to  fear  to  be  unable  to  satisfy  their  needs. 
As  far  as  games  are  immoral,  so  far  these  institutions 


156      A  PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 

are  advantageous  to  customs  by  favoring  the  strongest 
bents  of  our  nature.  The  government  ought  then  to 
encourage  them  and  respect  them  in  the  vicissitudes  of 
public  fortune ;  since  the  hopes  which  they  present  look 
toward  a  distant  future,  they  are  able  to  prosper  only 
when  sheltered  from  all  inquietude  during  their  exist- 
ence. It  is  an  advantage  that  the  institution  of  a 
representative  government  assures  them. 

Let  us  say  a  word  about  loans.  It  is  clear  that  in 
order  to  borrow  perpetually  it  is  necessary  to  pay  each 
year  the  product  of  the  capital  by  the  rate  of  interest. 
But  one  may  wish  to  discharge  this  principal  in  equal 
payments  made  during  a  definite  number  of  years, 
payments  which  are  called  annuities  and  whose  value 
is  obtained  in  this  manner.  Each  annuity  in  order  to 
be  reduced  at  the  actual  moment  ought  to  be  divided 
by  a  power  of  unity  augmented  by  the  rate  of  interest 
equal  to  the  number  of  years  after  which  this  annuity 
ought  to  be  paid.  Forming  then  a  geometric  progres- 
sion whose  first  term  is  the  annuity  divided  by  unity 
augmented  by  the  rate  of  interest,  and  whose  last  term 
is  this  annuity  divided  by  the  same  quantity  raised  to 
a  power  equal  to  the  number  of  years  during  which  the 
payment  should  have  been  made,  the  sum  of  this  pro- 
gression will  be  equivalent  to  the  capital  borrowed, 
which  will  determine  the  value  of  the  annuity.  A 
sinking  fund  is  at  bottom  only  a  means  of  converting 
into  annuities  a  perpetual  rent  with  the  sole  difference 
that  in  the  case  of  a  loan  by  annuities  the  interest  is 
supposed  constant,  while  the  interest  of  funds  acquired 
by  the  sinking  fund  is  variable.  If  it  were  the  same  in 
both  cases,  the  annuity  corresponding  to  the  funds 


INSTITUTIONS  BASED  UPON  PROBABILITIES.       157 

acquired  would  be  formed  by  these  funds  and  from 
this  annuity  the  State  contributes  annually  to  the  sink- 
ing fund. 

If  one  wishes  to  make  a  life  loan  it  will  be  observed 
that  the  tables  of  life  annuities  give  the  capital  required 
to  constitute  a  life  annuity  at  any  age,  a  simple  pro- 
portion will  give  the  rent  which  one  ought  to  pay  to 
the  individual  from  whom  the  capital  is  borrowed. 
From  these  principles  all  the  possible  kinds  of  loans 
may  be  calculated. 

The  principles  which  we  have  just  expounded  con- 
cerning the  benefits  and  the  losses  of  institutions  may 
serve  to  determine  the  mean  result  of  any  number  of 
observations  already  made,  when  one  wishes  to  regard 
the  deviations  of  the  results  corresponding  to  divers 
observations.  Let  us  designate  by  x  the  correction  of 
the  least  result  and  by  x  augmented  successively  by 
g,  q ',  q" ,  etc.,  the  corrections  of  the  following  results. 
Let  us  name  e,  e' ,  e" ,  etc.,  the  errors  of  the  observa- 
tions whose  law  of  probability  we  will  suppose  known. 
Each  observation  being  a  function  of  the  result,  it  is 
easy  to  see  that  by  supposing  the  correction  x  of  this 
result  to  be  very  small,  the  error  e  of  the  first  observa- 
tion wrill  be  equal  to  the  product  of  x  by  a  determined 
coefficient.  Likewise  the  error  e'  of  the  second  obser- 
vation will  be  the  product  of  the  sum  q  plus  x,  by  a 
determined  coefficient,  and  so  on.  The  probability  of 
the  error  e  being  given  by  a  known  function,  it  will  be 
expressed  by  the  same  function  of  the  first  of  the  pre- 
ceding products.  The  probability  of  e'  will  be  expressed 
by  the  same  function  of  the  second  of  these  products, 
and  so  on  of  the  others.  The  probability  of  the  simul- 


158      A  PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

taneous  existence  of  the  errors  e,  f',  c" ,  etc.,  will  be 
then  proportional  to  the  product  of  these  divers  func- 
tions, a  product  which  will  be  a  function  of  x.  This 
being  granted,  if  one  conceives  a  curve  whose  abscissa 
is  x,  and  whose  corresponding  ordinate  is  this  product, 
this  curve  will  represent  the  probability  of  the  divers 
values  of  x,  whose  limits  will  be  determined  by  the 
limits  of  the  errors  e,  e' ',  e" ,  etc.  Now  let  us  designate 
by  X  the  abscissa  which  it  is  necessary  to  choose ;  X 
diminished  by  x  will  be  the  error  which  would  be  com- 
mitted if  the  abscissa  x  were  the  true  correction.  This 
error,  multiplied  by  the  probability  of  x  or  by  the 
corresponding  ordinate  of  the  curve,  will  be  the  product 
of  the  loss  by  its  probability,  regarding,  as  one  should, 
this  error  as  a  loss  attached  to  the  choice  X.  Multi- 
plying this  product  by  the  differential  of  x  the  integral 
taken  from  the  first  extremity  of  the  curve  to  X  will 
be  the  disadvantage  of  X  resulting  from  the  values  of 
x  inferior  to  X,  For  the  values  of  x  superior  to  X,  x 
less  X  would  be  the  error  of  X  if  x  were  the  true  cor- 
rection ;  the  integral  of  the  product  of  x  by  the  corre- 
sponding ordinate  of  the  curve  and  by  the  differential 
of  x  will  be  then  the  disadvantage  of  X  resulting  from 
the  values  x  superior  to  x,  this  integral  being  taken 
from  x  equal  to  X  up  to  the  last  extremity  of  the 
curve.  Adding  this  disadvantage  to  the  preceding 
one,  the  sum  will  be  the  disadvantage  attached  to  the 
choice  of  X.  This  choice  ought  to  be  determined  by 
the  condition  that  this  disadvantage  be  a  minimum; 
and  a  very  simple  calculation  shows  that  for  this,  X 
ought  to  be  the  abscissa  whose  ordinate  divides  the 
curve  into  two  equal  parts,  so  that  it  is  thus  probable 


INSTITUTIONS   BASED   UPON  PROBABILITIES.       159 

that  the  true  value  of  x  falls  on  neither  the  one  side 
nor  the  other  of  X. 

Celebrated  geometricians  have  chosen  for  X  the 
most  probable  value  of  x  and  consequently  that  which 
corresponds  to  the  largest  ordinate  of  the  curve;  but 
the  preceding  value  appears  to  me  evidently  that  which 
the  theory  of  probability  indicates. 


CHAPTER   XVI. 

CONCERNING   ILLUSIONS  IN  THE  ESTIMATION 
OF  PROBABILITIES. 

THE  mind  has  its  illusions  as  the  sense  of  .sight;  and 
in  the  same  manner  that  the  sense  of  feeling  corrects 
the  latter,  reflection  and  calculation  correct  the  former. 
Probability  based  upon  a  daily  experience,  or  exag- 
gerated by  fear  and  by  hope,  strikes  us  more  than  a 
superior  probability  but  it  is  only  a  simple  result  of 
calculus.  Thus  we  do  not  fear  in  return  for  small 
advantages  to  expose  our  life  to  dangers  much  less 
improbable  than  the  drawing  of  a  quint  in  the  lottery 
of  France;  and  yet  no  one  would  wish  to  procure  for 
himself  the  same  advantages  with  the  certainty  of  losing 
his  life  if  this  quint  should  be  drawn. 

Our  passions,  our  prejudices,  and  dominating 
opinions,  by  exaggerating  the  probabilities  which  are 
favorable  to  them  and  by  attenuating  the  contrary 
probabilities,  are  the  abundant  sources  of  dangerous 
illusions. 

Present  evils  and  the  cause  which  produced  them 
effect  us  much  more  than  the  remembrance  of  evils 
produced  by  the  contrary  cause ;  they  prevent  us  from 

160 


ILLUSIONS  IN  THE  ESTIMATION  OF  PROBABILITIES.   161 

appreciating  with  justice  the  inconveniences  of  the  ones 
and  the  others,  and  the  probability  of  the  proper  means 
to  guard  ourselves  against  them.  It  is  this  which  leads 
alternately  to  despotism  and  to  anarchy  the  people 
who  are  driven  from  the  state  of  repose  to  which  they 
never  return  except  after  long  and  cruel  agitations. 

This  vivid  impression  which  we  receive  from  the 
presence  of  events,  and  which  allows  us  scarcely  to 
remark  the  contrary  events  observed  by  others,  is  a 
principal  cause  of  error  against  which  one  cannot  suffi- 
ciently guard  himself. 

It  is  principally  at  games  of  chance  that  a  multitude 
of  illusions  support  hope  and  sustain  it  against  unfavor- 
able chances.  The  majority  of  those  who  play  at 
lotteries  do  not  know  how  many  chances  are  to  their 
advantage,  how  many  are  contrary  to  them.  They 
see  only  the  possibility  by  a  small  stake  of  gaining  a 
considerable  sum,  and  the  projects  which  their  imagi- 
nation brings  forth,  exaggerate  to  their  eyes  the 
probability  of  obtaining  it;  the  poor  man  especially, 
excited  by  the  desire  of  a  better  fate,  risks  at  play  his 
necessities  by  clinging  to  the  most  unfavorable  com- 
binations which  promise  him  a  great  benefit.  All 
would  be  without  doubt  surprised  by  the  immense 
number  of  stakes  lost  if  they  could  know  of  them ;  but 
one  takes  care  on  the  contrary  to  give  to  the  winnings 
a  great  publicity,  which  becomes  a  new  cause  of  excite- 
ment for  this  funereal  play. 

When  a  number  in  the  lottery  of  France  has  not  been 
drawn  for  a  long  time  the  crowd  is  eager  to  cover  it 
with  stakes.  They  judge  since  the  number  has  not 
been  drawn  for  a  long  time  that  it  ought  at  the  next 


162      A  PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 

drawing  to  be  drawn  in  preference  to  others.  So 
common  an  error  appears  to  me  to  rest  upon  an  illusion 
by  which  one  is  carried  back  involuntarily  to  the  origin 
of  events.  It  is,  for  example,  very  improbable  that 
at  the  play  of  heads  and  tails  one  will  throw  heads  ten 
times  in  succession.  This  improbability  which  strikes 
us  indeed  when  it  has  happened  nine  times,  leads  us 
to  believe  that  at  the  tenth  throw  tails  will  be  thrown. 
But  the  past  indicating  in  the  coin  a  greater  propensity 
for  heads  than  for  tails  renders  the  first  of  the  events 
more  probable  than  the  second ;  it  increases  as  one  has 
seen  the  probability  of  throwing  heads  at  the  following 
throw.  A  similar  illusion  persuades  many  people  that 
one  can  certainly  win  in  a  lottery  by  placing  each  time 
upon  the  same  number,  until  it  is  drawn,  a  stake  whose 
product  surpasses  the  sum  of  all  the  stakes.  But  even 
when  similar  speculations  would  not  often  be  stopped 
by  the  impossibility  of  sustaining  them  they  would  not 
diminish  the  mathematical  disadvantage  of  speculators 
and  they  would  increase  their  moral  disadvantage, 
since  at  each  drawing  they  would  risk  a  very  large  part 
of  their  fortune. 

I  have  seen  men,  ardently  desirous  of  having  a  son, 
who  could  learn  only  with  anxiety  of  the  births  of  boys 
in  the  month  when  they  expected  to  become  fathers. 
Imagining  that  the  ratio  of  these  births  to  those  of  girls 
ought  to  be  the  same  at  the  end  of  each  month,  they 
judged  that  the  boys  already  born  would  render  more 
probable  the  births  next  of  girls.  Thus  the  extraction 
of  a  white  ball  from  an  urn  which  contains  a  limited 
number  of  white  balls  and  of  black  balls  increases  the 
probability  of  extracting  a  black  ball  at  the  following 


ILLUSIONS  IN   THE  ESTIMATION  OF  PROBABILITIES.   163 

drawing.  But  this  ceases  to  take  place  'when  the 
number  of  balls  in  the  urn  is  unlimited,  as  one  must 
suppose  in  order  to  compare  this  case  with  that  of 
births.  If,  in  the  course  of  a  month,  there  were  born 
many  more  boys  than  girls,  one  might  suspect  that 
toward  the  time  of  their  conception  a  general  cause 
had  favored  masculine  conception,  which  would  render 
more  probable  the  birth  next  of  a  boy.  The  irregular 
events  of  nature  are  not  exactly  comparable  to  the 
drawing  of  the  numbers  of  a  lottery  in  which  all  the 
numbers  are  mixed  at  each  drawing  in  such  a  manner 
as  to  render  the  chances  of  their  drawing  perfectly 
equal.  The  frequency  of  one  of  these  events  seems  to 
indicate  a  cause  slightly  favoring  it,  which  increases 
the  probability  of  its  next  return,  and  its  repetition 
prolonged  for  a  long  time,  such  as  a  long  series  of  rainy 
days,  may  develop  unknown  causes  for  its  change;  so 
that  at  each  expected  event  we  are  not,  as  at  each 
drawing  of  a  lottery,  led  back  to  the  same  state  of 
indecision  in  regard  to  what  ought  to  happen.  But  in 
proportion  as  the  observation  of  these  events  is  mul- 
tiplied, the  comparison  of  their  results  with  those  of 
lotteries  becomes  more  exact. 

By  an  illusion  contrary  to  the  preceding  ones  one 
seeks  in  the  past  drawings  of  the  lottery  of  France  the 
numbers  most  often  drawn,  in  order  to  form  combina- 
tions upon  which  one  thinks  to  place  the  stake  to 
advantage.  But  when  the  manner  in  which  the  mixing 
of  the  numbers  in  this  lottery  is  considered,  the  past 
ought  to  have  no  influence  upon  the  future.  The  very 
frequent  drawings  of  a  number  are  only  the  anomalies 
of  chance;  I  have  submitted  several  of  them  to  calcula- 


164     A  PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

tion  and  have  constantly  found  that  they  are  included 
within  the  limits  which  the  supposition  of  an  equal 
possibility  of  the  drawing  of  all  the  numbers  allows  us 
to  admit  without  improbability. 

In  a  long  series  of  events  of  the  same  kind  the  single 
chances  of  hazard  ought  sometimes  to  offer  the  singular 
veins  of  good  luck  or  bad  luck  which  the  majority  of 
players  do  not  fail  to  attribute  to  a  kind  of  fatality.  It 
happens  often  in  games  which  depend  at  the  same  time 
upon  hazard  and  upon  the  competency  of  the  players, 
that  that  one  who  loses,  troubled  by  his  loss,  seeks  to 
repair  it  by  hazardous  throws  which  he  would  shun  in 
another  situation ;  thus  he  aggravates  his  own  ill  luck 
and  prolongs  its  duration.  It  is  then  that  prudence 
becomes  necessary  and  that  it  is  of  importance  to  con- 
vince oneself  that  the  moral  disadvantage  attached  to 
unfavorable  chances  is  increased  by  the  ill  luck  itself. 

The  opinion  that  man  has  long  been  placed  in  the 
centre  of  the  universe,  considering  himself  the  special 
object  of  the  cares  of  nature,  leads  each  individual  to 
make  himself  the  centre  of  a  more  or  less  extended 
sphere  and  to  believe  that  hazard  has  preference  for 
him.  Sustained  by  this  belief,  players  often  risk  con- 
siderable sums  at  games  when  they  know  that  the 
chances  are  unfavorable.  In  the  conduct  of  life  a 
similar  opinion  may  sometimes  have  advantages ;  but 
most  often  it  leads  to  disastrous  enterprises.  Here  as 
everywhere  illusions  are  dangerous  and  truth  alone  is 
generally  useful. 

One  of  the  great  advantages  of  the  calculus  of  prob- 
abilities is  to  teach  us  to  distrust  first  opinions.  As  we 
recognize  that  they  often  deceive  when  they  may  be 


ILLUSIONS  IN  THE  ESTIMATION  OF  PROBABILITIES.  165 

submitted  to  calculus,  we  ought  to  conclude  that  in 
other  matters  confidence  should  be  given  only  after 
extreme  circumspection.  Let  us  prove  this  by  example. 

An  urn  contains  four  balls,  black  and  white,  but  which 
are  not  all  of  the  same  color.  One  of  these  balls  has 
been  drawn  whose  color  is  white  and  which  has  been 
put  back  in  the  urn  in  order  to  proceed  again  to  similar 
drawings.  One  demands  the  probability  of  extracting 
only  black  balls  in  the  four  following  drawings. 

If  the  white  and  black  were  in  equal  number  this 
probability  would  be  the  fourth  power  of  the  probability 
£  of  extracting  a  black  ball  at  each  drawing ;  it  would 
be  then  T^.  But  the  extraction  of  a  white  ball  at  the 
first  drawing  indicates  a  superiority  in  the  number  of 
white  balls  in  the  urn ;  for  if  one  supposes  in  the  urn 
three  white  balls  and  one  black  the  probability  of 
extracting  a  white  ball  is  |;  it  is  £  if  one  supposes  two 
white  balls  and  two  black;  finally  it  is  reduced  to  J  if 
one  supposes  three  black  balls  and  one  white.  Follow- 
ing the  principle  of  the  probability  of  causes  drawn 
from  events  the  probabilities  of  these  three  suppositions 
are  among  themselves  as  the  quantities  £,  f,  £;  they 
are  consequently  equal  to  |,  f,  £.  It  is  thus  a  bet  of 
5  against  i  that  the  number  of  black  balls  is  inferior, 
or  at  the  most  equal,  to  that  of  the  white.  It  seems 
then  that  after  the  extraction  of  a  white  ball  at  the  first 
drawing,  the  probability  of  extracting  successively  four 
black  balls  ought  to  be  less  than  in  the  case  of  the 
equality  of  the  colors  or  smaller  than  one  sixteenth. 
However,  it  is  not,  and  it  is  found  by  a  very  simple 
calculation  that  this  probability  is  greater  than  one 
fourteenth.  Indeed  it  would  be  the  fourth  power 


1 66     A  PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 

of  £,  of  |,  and  of  |  in  the  first,  the  second,  and  the 
third  of  the  preceding  suppositions  concerning  the 
colors  of  the  balls  in  the  urn.  Multiplying  respectively 
each  power  by  the  probability  of  the  corresponding 
supposition,  or  by  f ,  f ,  and  £,  the  sum  of  the  products 
will  be  the  probability  of  extracting  successively  four 
black  balls.  One  has  thus  for  this  probability  ^2¥9¥,  a 
fraction  greater  than  -fa.  This  paradox  is  explained 
by  considering  that  the  indication  of  the  superiority  of 
white  balls  over  the  black  ones  at  the  first  drawing 
does  not  exclude  at  all  the  superiority  of  the  black  balls 
over  the  white  ones,  a  superiority  which  excludes  the 
supposition  of  the  equality  of  the  colors.  But  this 
superiority,  though  but  slightly  probable,  ought  to 
render  the  probability  of  drawing  successively  a  given 
number  of  black  balls  greater  than  in  this  supposition 
if  the  number  is  considerable ;  and  one  has  just  seen 
that  this  commences  when  the  given  number  is  equal 
to  four.  Let  us  consider  again  an  urn  which  contains 
several  white  and  black  balls.  Let  us  suppose  at  first 
that  there  is  only  one  white  ball  and  one  black.  It  is 
then  an  even  bet  that  a  white  ball  will  be  extracted  in 
one  drawing.  But  it  seems  for  the  equality  of  the  bet 
that  one  who  bets  on  extracting  the  white  ball  ought 
to  have  two  drawings  if  the  urn  contains  two  black 
and  one  white,  three  drawings  if  it  contains  three  black 
and  one  white,  and  so  on ;  it  is  supposed  that  after  each 
drawing  the  extracted  ball  is  placed  again  in  the  urn. 

We  are  convinced  easily  that  this  first  idea  is 
erroneous.  Indeed  in  the  case  of  two  black  and  one 
white  ball,  the  probability  of  extracting  two  black  in 
two  drawings  is  the  second  power  of  f  or  ^ ;  but  this 


ILLUSIONS  IN   THE  ESTIMATION  OF  PROBABILITIES.   167 

probability  added  to  that  of  drawing  a  white  ball  in  two 
drawings  is  certainty  or  unity,  since  it  is  certain  that 
two  black  balls  or  at  least  one  white  ball  ought  to  be 
drawn ;  the  probability  in  this  last  case  is  then  -|,  a 
fraction  greater  than  f .  There  would  still  be  a  greater 
advantage  in  the  bet  of  drawing  one  white  ball  in  five 
draws  when  the  urn  contains  five  black  and  one  white 
ball ;  this  bet  is  even  advantageous  in  four  drawings ; 
it  returns  then  to  that  of  throwing  six  in  four  throws 
with  a  single  die. 

The  Chevalier  de  Mere,  who  caused  the  invention 
of  the  calculus  of  probabilities  by  encouraging  his  friend 
Pascal,  the  great  geometrician,  to  occupy  himself  with 
it,  said  to  him  ' '  that  he  had  found  error  in  the  num- 
bers by  this  ratio.  If  we  undertake  to  make  six  with 
one  die  there  is  an  advantage  in  undertaking  it  in  four 
throws,  as  671  to  625.  If  we  undertake  to  make  two 
sixes  with  two  dice,  there  is  a  disadvantage  in  under- 
taking in  24  throws.  At  least  24  is  to  36,  the  number 
of  the  faces  of  the  two  dice,  as  4  is  to  6,  the  number 
of  faces  of  one  die."  "This  was,"  wrote  Pascal  to 
Fermat,  ' '  his  great  scandal  which  caused  him  to  say 
boldly  that  the  propositions  were  not  constant  and  that 
arithmetic  was  demented.  .  .  .  He  has  a  very  good 
mind,  but  he  is  not  a  geometrician,  which  is,  as  you 
know,  a  great  fault. ' '  The  Chevalier  de  Mere,  deceived 
by  a  false  analogy,  thought  that  in  the  case  of  the 
equality  of  bets  the  number  of  throws  ought  to  increase 
in  proportion  to  the  number  of  all  the  chances  possible, 
which  is  not  exact,  but  which  approaches  exactness  as 
this  number  becomes  larger. 

One  has  endeavored  to  explain  the  superiority  of  the 


1 68      A  PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 

births  of  boys  over  those  of  girls  by  the  general  desire 
of  fathers  to  have  a  son  who  would  perpetuate  the 
name.  Thus  by  imagining  an  urn  filled  with  an  infinity 
of  white  and  black  balls  in  equal  number,  and  suppos- 
ing a  great  number  of  persons  each  of  whom  draws  a 
ball  from  this  urn  and  continues  with  the  intention  of 
stopping  when  he  shall  have  extracted  a  white  ball, 
one  has  believed  that  this  intention  ought  to  render  the 
number  of  white  balls  extracted  superior  to  that  of  the 
black  ones.  Indeed  this  intention  gives  necessarily 
after  all  the  drawings  a  number  of  white  balls  equal 
to  that  of  persons,  and  it  is  possible  that  these  draw- 
ings would  never  lead  a  black  ball.  But  it  is  easy  to 
see  that  this  first  notion  is  only  an  illusion;  for  if  one 
conceives  that  in  the  first  drawing  all  the  persons  draw 
at  once  a  ball  from  the  urn,  it  is  evident  that  their 
intention  can  have  no  influence  upon  the  color  of  the 
balls  which  ought  to  appear  at  this  drawing.  Its 
unique  effect  will  be  to  exclude  from  the  second  draw- 
ing the  persons  who  shall  have  drawn  a  white  one  at 
the  first.  It  is  likewise  apparent  that  the  intention  of 
the  persons  who  shall  take  part  in  the  new  drawing 
will  have  no  influence  upon  the  color  of  the  balls  which 
shall  be  drawn,  and  that  it  will  be  the  same  at  the  fol- 
lowing drawings.  This  intention  will  have  no  influence 
then  upon  the  color  of  the  balls  extracted  in  the  totality 
of  drawings ;  it  will,  however,  cause  more  or  fewer  to 
participate  at  each  drawing.  The  ratio  of  the  white 
balls  extracted  to  the  black  ones  will  differ  thus  very 
little  from  unity.  It  follows  that  the  number  of  persons 
being  supposed  very  large,  if  observation  gives  between 
the  colors  extracted  a  ratio  which  differs  sensibly  from 


ILLUSIONS  IN   THE  ESTIMATION  OF  PROBABILITIES.   169 

unity,  it  is  very  probable  that  the  same  difference  is 
found  between  unity  and  the  ratio  of  the  white  balls  to 
the  black  contained  in  the  urn. 

I  count  again  among  illusions  the  application  which 
Liebnitz  and  Daniel  Bernoulli  have  made  of  the  cal- 
culus of  probabilities  to  the  summation  of  series.  If 
one  reduces  the  fraction  whose  numerator  is  unity  and 
whose  denominator  is  unity  plus  a  variable,  in  a  series 
prescribed  by  the  ratio  to  the  powers  of  this  variable,  it 
is  easy  to  see  that  in  supposing  the  variable  equal  to 
unity  the  fraction  becomes  £,  and  the  series  becomes 
plus  one,  minus  one,  plus  one,  minus  one,  etc.  In 
adding  the  first  two  terms,  the  second  two,  and  so  on, 
the  series  is  transformed  into  another  of  which  each 
term  is  zero.  Grandi,  an  Italian  Jesuit,  concluded 
from  this  the  possibility  of  the  creation ;  because  the 
series  being  always  £,  he  saw  this  fraction  spring  from 
an  infinity  of  zeros  or  from  nothing.  It  was  thus  that 
Liebnitz  believed  he  saw  the  image  of  creation  in  his 
binary  arithmetic  where  he  employed  only  the  two 
characters,  unity  and  zero.  He  imagined,  since  God 
can  be  represented  by  unity  and  nothing  by  zero,  that 
the  Supreme  Being  had  drawn  from  nothing  all  beings, 
as  unity  with  zero  expresses  all  the  numbers  in  this 
system  of  arithmetic.  This  idea  was  so  pleasing  to 
Liebnitz  that  he  communicated  it  to  the  Jesuit 
Grimaldi,  president  of  the  tribunal  of  methematics  in 
China,  in  the  hope  that  this  emblem  of  creation  would 
convert  to  Christianity  the  emperor  there  who  particu- 
larly loved  the  sciences.  I  report  this  incident  only 
to  show  to  what  extent  the  prejudices  of  infancy  can 
mislead  the  greatest  men. 


17°      A  PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

Liebnitz,  always  led  by  a  singular  and  very  loose 
metaphysics,  considered  that  the  series  plus  one,  minus 
one,  plus  one,  etc.,  becomes  unity  or  zero  according  as 
one  stops  at  a  number  of  terms  odd  or  even ;  and  as  in 
infinity  there  is  no  reason  to  prefer  the  even  number 
to  the  odd,  one  ought  following  the  rules  of  probability, 
to  take  the  half  of  the  results  relative  to  these  two  kinds 
of  numbers,  and  which  are  zero  and  unity,  which  gives 
£  for  the  value  of  the  series.  Daniel  Bernoulli  has 
since  extended  this  reasoning  to  the  summation  of 
series  formed  from  periodic  terms.  But  all  these  series 
have  no  values  properly  speaking ;  they  get  them  only 
in  the  case  where  their  terms  are  multiplied  by  the 
successive  powers  of  a  variable  less  than  unity.  Then 
these  series  are  always  convergent,  however  small  one 
supposes  the  difference  of  the  variable  from  unity;  and 
it  is  easy  to  demonstrate  that  the  values  assigned  by 
Bernoulli,  by  virtue  of  the  rule  of  probabilities,  are  the 
same  values  of  the  generative  fraction  of  the  series, 
when  one  supposes  in  these  fractions  the  variable  equal 
to  unity.  These  values  are  again  the  limits  which  the 
series  approach  more  and  more,  in  proportion  as  the 
variable  approaches  unity.  But  when  the  variable  is 
exactly  equal  to  unity  the  series  cease  to  be  convergent ; 
they  have  values  only  as  far  as  one  arrests  them.  The 
remarkable  ratio  of  this  application  of  the  calculus  of 
probabilities  with  the  limits  of  the  values  of  periodic 
series  supposes  that  the  terms  of  these  series  are  multi- 
plied by  all  the  consecutive  powers  of  the  variable. 
But  this  series  may  result  from  the  development  of  an 
infinity  of  different  fractions  in  which  this  did  not  occur. 
Thus  the  series  plus  one,  minus -one,  plus  one,  etc., 


ILLUSIONS  IN   THE  ESTIMATION  OF  PROBABILITIES.   i?i 

may  spring  from  the  development  of  a  fraction  whose 
numerator  is  unity  plus  the  variable,  and  whose 
denominator  is  this  numerator  augmented  by  the  square 
of  the  variable.  Supposing  the  variable  equal  to  unity, 
this  development  changes,  in  the  series  proposed,  and 
the  generative  fraction  becomes  equal  to  f ;  the  rules 
of  probabilities  would  give  then  a  false  result,  which 
proves  how  dangerous  it  would  be  to  employ  similar 
reasoning,  especially  in  the  mathematical  sciences, 
which  ought  to  be  especially  distinguished  by  the  rigor 
of  their  operations. 

We  are  led  naturally  to  believe  that  the  order 
according  to  which  we  see  things  renewed  upon  the 
earth  has  existed  from  all  times  and  will  continue 
always.  Indeed  if  the  present  state  of  the  universe 
were  exactly  similar  to  the  anterior  state  which  has 
produced  it,  it  would  give  birth  in  its  turn  to  a  similar 
state;  the  succession  of  these  states  would  then  be 
eternal.  I  have  found  by  the  application  of  analysis  to 
the  law  of  universal  gravity  that  the  movement  of  rota- 
tion and  of  revolution  of  the  planets  and  satellites,  and 
the  position  of  the  orbits  and  of  their  equators  are  sub- 
jected only  to  periodic  inequalities.  In  comparing  With 
ancient  eclipses  the  theory  of  the  secular  equation  of 
the  moon  I  have  found  that  since  Hipparchus  the 
duration  of  the  day  has  not  varied  by  the  hundredth  of 
a  second,  and  that  the  mean  temperature  of  the  earth 
has  not  diminished  the  one-hundredth  of  a  degree. 
Thus  the  stability  of  actual  order  appears  established 
at  the  same  time  by  theory  and  by  observations.  But 
this  order  is  effected  by  divers  causes  which  an  atten- 


172      A  PHILOSOPHICAL   ESSAY  ON  PROBABILITIES, 

tive  examination  reveals,  and  which  it  is  impossible  to 
submit  to  calculus. 

The  actions  of  the  ocean,  of  the  atmosphere,  and  of 
meteors,  of  earthquakes,  and  the  eruptions  of  volcanoes, 
agitate  continually  the  surface  of  the  earth  and  ought 
to  effect  in  the  long  run  great  changes.  The  tempera- 
ture of  climates,  the  volume  of  the  atmosphere,  and  the 
proportion  of  the  gases  which  constitute  it,  may  vary  in 
an  inappreciable  manner.  The  instruments  and  the 
means  suitable  to  determine  these  variations  being 
new,  observation  has  been  unable  up  to  this  time  to 
teach  us  anything  in  this  regard.  But  it  is  hardly 
probable  that  the  causes  which  absorb  and  renew  the 
gases  constituting  the  air  maintain  exactly  their  respec- 
tive proportions.  A  long  series  of  centuries  will  show 
the  alterations  which  are  experienced  by  all  these 
elements  so  essential  to  the  conservation  of  organized 
beings.  Although  historical  monuments  do  not  go 
back  to  a  very  great  antiquity  they  offer  us  nevertheless 
sufficiently  great  changes  which  have  come  about  by 
the  slow  and  continued  action  of  natural  agents. 
Searching  in  the  bowels  of  the  earth  one  discovers 
numerous  debris  of  former  nature,  entirely  different 
from  the  present.  Moreover,  if  the  entire  earth  was  in 
the  beginning  fluid,  as  everything  appears  to  indicate, 
one  imagines  that  in  passing  from  that  state  to  the  one 
which  it  has  now,  its  surface  ought  to  have  experienced 
prodigious  changes.  The  heavens  itself  in  spite  of  the 
order  of  its  movements,  is  not  unchangeable.  The 
resistance  of  light  and  of  other  ethereal  fluids,  and  the 
attraction  of  the  stars  ought,  after  a  great  number  of 
centuries,  to  alter  considerably  the  planetary  move- 


ILLUSIONS  IN   THE  ESTIMATION  OF  PROBABILITIES.    173 

ments.  The  variations  already  observed  in  the  stars 
and  in  the  form  of  the  nebulae  give  us  a  presentiment 
of  those  which  time  will  develop  in  the  system  of  these 
great  bodies.  One  may  represent  the  successive  states 
of  the  universe  by  a  curve,  of  which  time  would  be  the 
abscissa  and  of  which  the  ordinates  are  the  divers 
states.  Scarcely  knowing  an  element  of  this  curve  we 
are  far  from  being  able  to  go  back  to  its  origin ;  and 
if  in  order  to  satisfy  the  imagination,  always  restless 
from  our  ignorance  of  the  cause  of  the  phenomena 
which  interest  it,  one  ventures  some  conjectures  it  is 
wise  to  present  them  only  with  extreme  reserve. 

There  exists  in  the  estimation  of  probabilities  a  kind 
of  illusions,  which  depending  especially  upon  the  laws 
of  the  intellectual  organization  demands,  in  order  to 
secure  oneself  against  them,  a  profound  examination 
of  these  laws.  The  desire  to  penetrate  into  the  future 
and  the  ratios  of  some  remarkable  events,  to  the  predic- 
tions of  astrologers,  of  diviners  and  soothsayers,  to 
presentiments  and  dreams,  to  the  numbers  and  the 
days  reputed  lucky  or  unlucky,  have  given  birth  to  a 
multitude  of  prejudices  still  very  widespread.  One 
does  not  reflect  upon  the  great  number  of  non-coinci- 
dences which  have  made  no  impression  or  which  are 
unknown.  However,  it  is  necessary  to  be  acquainted 
with  them  in  order  to  appreciate  the  probability  of  the 
causes  to  which  the  coincidences  are  attributed.  This 
knowledge  would  confirm  without  doubt  that  which 
reason  tells  us  in  regard  to  these  prejudices.  Thus  the 
philosopher  of  antiquity  to  whom  is  shown  in  a  temple, 
in  order  to  exalt  the  power  of  the  god  who  is  adored 
there,  the  ex  veto  of  all  those  who  after  having  invoked 


174      A  PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

it  were  saved  from  shipwreck,  presents  an  incident 
consonant  with  the  calculus  of  probabilities,  observing 
that  he  does  not  see  inscribed  the  names  of  those  who, 
in  spite  of  this  invocation,  have  perished.  Cicero  has 
refuted  all  these  prejudices  with  much  reason  and 
eloquence  in  his  Treatise  on  Divination,  which  he  ends 
by  a  passage  which  I  shall  cite ;  for  on.e  loves  to  find 
again  among  the  ancients  the  thunderbolts  of  reason, 
which,  after  having  dissipated  all  the  prejudices  by  its 
light,  shall  become  the  sole  foundation  of  human  insti- 
tutions. 

"It  is  necessary,"  says  the  Roman  orator,  "to 
reject  divination  by  dreams  and  all  similar  prejudices. 
Widespread  superstition  has  subjugated  the  majority 
of  minds  and  has  taken  possession  of  the  feebleness  of 
men.  It  is  this  we  have  expounded  in  our  books  upon 
the  nature  of  the  gods  and  especially  in  this  work, 
persuaded  that  we  shall  render  a  service  to  others  and 
to  ourselves  if  we  succeed  in  destroying  superstition. 
However  (and  I  desire  especially  in  this  regard  my 
thought  be  well  comprehended),  in  destroying  super- 
stition I  am  far  from  wishing  to  disturb  religion. 
Wisdom  enjoins  us  to  maintain  the  institutions  and  the 
ceremonies  of  our  ancestors,  touching  the  cult  of  the 
gods.  Moreover,  the  beauty  of  the  universe  and  the 
order  of  celestial  things  force  us  to  recognize  some 
superior  nature  which  ought  to  be  remarked  and 
admired  by  the  human  race.  But  as  far  as  it  is  proper 
to  propagate  religion,  which  is  joined  to  the  knowledge 
of  nature,  so  far  it  is  necessary  to  work  toward  the 
extirpation  of  superstition,  for  it  torments  one,  impor- 
tunes one,  and  pursues  one  continually  and  in  all  places. 


ILLUSIONS  IN   THE  ESTIMATION  OF  PROBABILITIES.   i?S 

If  one  consult  a  diviner  or  a  soothsayer,  if  one  immo- 
lates a  victim,  if  one  regards  the  flight  of  a  bird,  if  one 
encounters  a  Chaldean  or  an  aruspex,  if  it  lightens,  if 
it  thunders,  if  the  thunderbolt  strikes,  finally,  if  there 
is  born  or  is  manifested  a  kind  of  prodigy,  things  one 
of  which  ought  often  to  happen,  then  superstition 
dominates  and  leaves  no  repose.  Sleep  itself,  this 
refuge  of  mortals  in  their  troubles  and  their  labors, 
becomes  by  it  a  new  source  of  inquietude  and  fear. ' ' 

All  these  prejudices  and  the  terrors  which  they 
inspire  are  connected  with  physiological  causes  which 
continue  sometimes  to  operate  strongly  after  reason 
has  disabused  us  of  them.  But  the  repetition  of  acts 
contrary  to  these  prejudices  can  always  destroy  them. 


CHAPTER   XVII. 

CONCERNING    THE  VARIOUS  MEANS  OF 
APPROACHING  CERTAINTY. 

INDUCTION,  analogy,  hypotheses  founded  upon  facts 
and  rectified  continually  by  new  observations,  a  happy 
tact  given  by  nature  and  strengthened  by  numerous 
comparisons  of  its  indications  with  experience,  such 
are  the  principal  means  for  arriving  at  truth. 

If  one  considers  a  series  of  objects  of  the  same 
nature  one  perceives  among  them  and  in  their  changes 
ratios  which  manifest  themselves  more  and  more  in 
proportion  as  the  series  is  prolonged,  and  which, 
extending  and  generalizing  continually,  lead  finally  to 
the  principle  from  which  they  were  derived.  But  these 
ratios  are  enveloped  by  so  many  strange  circumstances 
that  it  requires  great  sagacity  to  disentangle  them  and 
to  recur  to  this  principle:  it  is  in  this  that  the  true 
genius  of  sciences  consists.  Analysis  and  natural 
philosophy  owe  their  most  important  discoveries  to  this 
fruitful  means,  which  is  called  inditction.  Newton  was 
indebted  to  it  for  his  theorem  of  the  binomial  and  the 
principle  of  universal  gravity.  It  is  difficult  to  appre- 
ciate the  probability  of  the  results  of  induction,  which  is 

176 


VARIOUS  MEANS   OF  APPROACHING   CERTAINTY.    i?7 

based  upon  this  that  the  simplest  ratios  are  the  most 
common ;  this  is  verified  in  the  formulae  of  analysis  and 
is  found  again  in  natural  phenomena,  in  crystallization, 
and  in  chemical  combinations.  This  simplicity  of 
ratios  will  not  appear  astonishing  if  we  consider  that 
all  the  effects  of  nature  are  only  mathematical  results 
of  a  small  number  of  immutable  laws. 

Yet  induction,  in  leading  to  the  discovery  of  the 
general  principles  of  the  sciences,  does  not  suffice  to 
establish  them  absolutely.  It  is  always  necessary  to 
confirm  them  by  demonstrations  or  by  decisive  experi- 
ences; for  the  history  of  the  sciences  shows  us  that 
induction  has  sometimes  led  to  inexact  results.  I  shall 
cite,  for  example,  a  theorem  of  Fermat  in  regard  to 
primary  numbers.  This  great  geometrician,  who  had 
meditated,  profoundly  upon  this  theorem,  sought  a 
formula  which,  containing  only  primary  numbers,  gave 
directly  a  primary  number  greater  than  any  other 
number  assignable.  Induction  led  him  to  think  that 
two,  raised  to  a  power  which  was  itself  a  power  of  two, 
formed  with  unity  a  primary  number.  Thus,  two 
raised  to  the  square  plus  one,  forms  the  primary  num- 
ber five;  two  raised  to  the  second  power  of  two,  or 
sixteen,  forms  with  one  the  primary  number  seventeen. 
He  found  that  this  was  still  true  for  the  eighth  and  the 
sixteenth  power  of  two  augmented  by  unity;  and  this 
induction,  based  upon  several  arithmetical  considera- 
tions, caused  him  to  regard  this  result  as  general. 
However,  he  avowed  that  he  had  not  demonstrated  it. 
Indeed,  Euler  recognized  that  this  does  not  hold  for 
the  thirty-second  power  of  two,  which,  augmented  by 
unity,  gives  4,294,967.297,  a  number  divisible  by  641. 


I?8     A  PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 

We  judge  by  induction  that  if  various  events,  move- 
ments, for  example,  appear  constantly  and  have  been 
long  connected  by  a  simple  ratio,  they  will  continue 
to  be  subjected  to  it;  and  we  conclude  from  this,  by 
the  theory  of  probabilities,  that  this  ratio  is  due,  not  to 
hazard,  but  to  a  regular  cause.  Thus  the  equality  of 
the  movements  of  the  rotation  and  the  revolution  of 
the  moon ;  that  of  the  movements  of  the  nodes  of  the 
orbit  and  of  the  lunar  equator,  and  the  coincidence  of 
these  nodes ;  the  singular  ratio  of  the  movements  of 
the  first  three  satellites  of  Jupiter,  according  to  which 
the  mean  longitude  of  the  first  satellite,  less  three  times 
that  of  the  second,  plus  two  times  that  of  the  third,  is 
equal  to  two  right  angles ;  the  equality  of  the  interval 
of  the  tides  to  that  of  the  passage  of  the  moon  to  the 
meridian ;  the  return  of  the  greatest  tides .  with  the 
syzygies,  and  of  the  smallest  with  the  quadratures ;  all 
these  things,  which  have  been  maintained  since  they 
were  first  observed,  indicate  with  an  extreme  prob- 
ability, the  existence  of  constant  causes  which  geome- 
tricians have  happily  succeeded  in  attaching  to  the  law 
of  universal  gravity,  and  the  knowledge  of  which 
renders  certain  the  perpetuity  of  these  ratios. 

The  chancellor  Bacon,  the  eloquent  promoter  of  the 
true  philosophical  method,  has  made  a  very  strange 
misuse  of  induction  in  order  to  prove  the  immobility  of 
the  earth.  He  reasons  thus  in  the  Novum  Organum, 
his  finest  work :  ' '  The  movement  of  the  stars  from 
the  orient  to  the  Occident  increases  in  swiftness,  in 
proportion  to  their  distance  from  the  earth.  This 
movement  is  swiftest  with  the  stars ;  it  slackens  a  little 
with  Saturn,  a  little  more  with  Jupiter,  and  so  on  to 


VARIOUS  MEANS  OF  APPROACHING  CERTAINT Y.  'i?9 

the  moon  and  the  highest  comets.  It  is  still  percepti- 
ble in  the  atmosphere,  especially  between  the  tropics, 
on  account  of  the  great  circles  which  the  molecules  of 
the  air  describe  there ;  finally,  it  is  almost  inappreciable 
with  the  ocean;  it  is  then  nil  for  the  earth."  But  this 
induction  proves  only  that  Saturn,  and  the  stars  which 
are  inferior  to  it,  have  their  own  movements,  contrary 
to  the  real  or  apparent  movement  which  sweeps  the 
whole  celestial  sphere  from  the  orient  to  the  Occident, 
and  that  these  movements  appear  slower  with  the  more 
remote  stars,  which  is  conformable  to  the  laws  of 
optics.  Bacon  ought  to  have  been  struck  by  the 
inconceivable  swiftness  which  the  stars  require  in  order 
to  accomplish  their  diurnal  revolution,  if  the  earth  is 
immovable,  and  by  the  extreme  simplicity  with  which 
its  rotation  explains  how  bodies  so  distant,  the  ones 
from  the  others,  as  the  stars,  the  sun,  the  planets,  and 
the  moon,  all  seem  subjected  to  this  revolution.  As 
to  the  ocean  and  to  the  atmosphere,  he  ought  not  to 
compare  their  movement  with  that  of  the  stars  which 
are  detached  from  the  earth ;  but  since  the  air  and  the 
sea  make  part  of  the  terrestrial  globe,  they  ought  to 
participate  in  its  movement  or  in  its  repose.  It  is 
singular  that  Bacon,  carried  to  great  prospects  by  his 
genius,  was  not  won  over  by  the  majestic  idea  which 
the  Copernican  system  of  the  universe  offers.  He  was 
able,  however,  to  find  in  favor  of  that  system,  strong 
analogies  in  the  discoveries  of  Galileo,  which  were 
continued  by  him.  He  has  given  for  the  search  after 
truth  the  precept,  but  not  the  example.  But  by 
insisting,  with  all  the  force  of  reason  and  of  eloquence, 
upon  the  necessity  of  abandoning  the  insignificant 


i8o      A  PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 

subtilities  of  the  school,  in  order  to  apply  oneself  to 
observations  and  to  experiences,  and  by  indicating  the 
true  method  of  ascending  to  the  general  causes  of 
phenomena,  this  great  philosopher  contributed  to  the 
immense  strides  which  the  human  mind  made  in  the 
grand  century  in  which  he  terminated  his  career. 

Analogy  is  based  upon  the  probability,  that  similar 
things  have  causes  of  the  same  kind  and  produce  the 
same  effects.  This  probability  increase  as  the  simili- 
tude becomes  more  perfect.  Thus  we  judge  without 
doubt  that  beings  provided  with  the  same  organs, 
doing  the  same  things,  experience  the  same  sensations, 
and  are  moved  by  the  same  desires.  The  probability 
that  the  animals  which  resemble  us  have  sensations 
analogous  to  ours,  although  a  little  inferior  to  that 
which  is  relative  to  individuals  of  our  species,  is  still 
exceedingly  great;  and  it  has  required  all  the  influence 
of  religious  prejudices  to  make  us  think  with  some 
philosophers  that  animals  are  mere  automatons.  The 
probability  of  the  existence  of  feeling  decreases  in  the 
same  proportion  as  the  similitude  of  the  organs  with 
ours  diminishes,  but  it  is  always  very  great,  even  with 
insects.  In  seeing  those  of  the  same  species  execute 
very  complicated  things  exactly  in  the  same  manner 
from  generation  to  generation,  and  without  having 
learned  them,  one  is  led  to  believe  that  they  act  by  a 
kind  of  affinity  analogous  to  that  which  brings  together 
the  molecules  of  crystals,  but  which,  together  with  the 
sensation  attached  to  all  animal  organization,  produces, 
with  the  regularity  of  chemical  combinations,  combina- 
tions that  are  much  more  singular;  one  might,  perhaps, 
name  this  mingling  of  elective  affinities  and  sensations 


VARIOUS  MEANS  OF  APPROACHING  CERTAINTY.  181 

animal  affinity.  Although  there  exists  a  great  analogy 
between  the  organization  of  plants  and  that  of  animals, 
it  does  not  seem  to  me  sufficient  to  extend  to  vegetables 
the  sense  of  feeling ;  but  nothing  authorizes  us  in  deny- 
ing it  to  them. 

Since  the  sun  brings  forth,  bythe  beneficent  action 
of  its  light  and  of  its  heat,  the  animals  and  plants 
Avhich  cover  the  earth,  we  judge  by  analogy  that  it 
produces  similar  effects  upon  the  other  planets ;  for  it 
is  not  natural  to  think  that  the  cause  whose  activity  we 
see  developed  in  so  many  ways  should  be  sterile  upon 
so  great  a  planet  as  Jupiter,  which,  like  the  terrestrial 
globe,  has  its  days,  its  nights,  and  its  years,  and  upon 
which  observations  indicate  changes  which  suppose 
very  active  forces.  Yet  this  would  be  giving  too  great 
an  extension  to  analogy  to  conclude  from  it  the  simili- 
tude of  the  inhabitants  of  the  planets  and  of  the  earth. 
Man,  made  for  the  temperature  which  he  enjoys,  and 
for  the  element  which  he  breathes,  would  not  be  able, 
according  to  all  appearance,  to  live  upon  the  other 
planets.  But  ought  there  not  to  be  an  infinity  of 
organization  relative  to  the  various  constitutions  of  the 
globes  of  this  universe  ?  If  the  single  difference  of  the 
elements  and  of  the  climates  make  so  much  variety  in 
terrestrial  productions,  how  much  greater  the  difference 
ought  to  be  among  those  of  the  various  planets  and  of 
their  satellites !  The  most  active  imagination  can  form 
no  idea  of  it ;  but  their  existence  is  very  probable. 

We  are  led  by  a  strong  analogy  to  regard  the  stars 
as  so  many  suns  endowed,  like  ours,  with  an  attractive 
power  proportional  to  the  mass  and  reciprocal  to  the 
square  of  the  distances ;  for  this  power  being  demon- 


1 82      A  PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

strated  for  all  the  bodies  of  the  solar  system,  and  for 
their  smallest  molecules,  it  appears  to  appertain  to  all 
matter.  Already  the  movements  of  the  small  stars, 
which  have  been  called  double,  on  account  of  their 
conjunction,  appear  to  indicate  it;  a  century  at  most  of 
precise  observations,  by  verifying  their  movements  of 
revolution,  the  ones  about  the  others,  will  place  beyond 
doubt  their  reciprocal  attractions. 

The  analogy  which  leads  us  to  make  each  star  the 
centre  of  a  planetary  system  is  far  less  strong  than 
the  preceding  one ;  but  it  acquires  probability  by  the 
hypothesis  which  has  been  proposed  in  regard  to 
the  formation  of  the  stars  and  of  the  sun;  for  in  this 
hypothesis  each  star,  having  been  like  the  sun,  primi- 
tively environed  by  a  vast  atmosphere,  it  is  natural  to 
attribute  to  this  atmosphere  the  same  effects  as  to  the 
solar  atmosphere,  and  to  suppose  that  it  has  produced, 
in  condensing,  planets  and  satellites. 

A  great  number  of  discoveries  in  the  sciences  is  due 
to  analogy.  I  shall  cite  as  one  of  the  most  remarkable, 
the  discovery  of  atmospheric  electricity,  to  which  one 
has  been  led  by  the  analogy  of  electric  phenomena 
with  the  effects  of  thunder. 

The  surest  method  which  can  guide  us  in  the  search 
for  truth,  consists  in  rising  by  induction  from  phenomena 
to  laws  and  from  laws  to  forces.  Laws  are  the  ratios 
which  connect  particular  phenomena  together:  when 
they  have  shown  the  general  principle  of  the  forces 
from  which  they  are  derived,  one  verifies  it  either  by 
direct  experiences,  when  this  is  possible,  or  by  exami- 
nation if  it  agrees  with  known  phenomena;  and  if  by 
a  rigorous  analysis  we  see  them  proceed  from  this 


YAR1OUS  MEJNS   OF  APPROACHING   CERTAINTY.  183 

principle,  even  in  their  small  details,  and  if,  moreover, 
they  are  quite  varied  and  very  numerous,  then  science 
acquires  the  highest  degree  of  certainty  and  of  perfec- 
tion that  it  is  able  to  attain.  Such,  astronomy  has 
become  by  the  discovery  of  universal  gravity.  The 
history  of  the  sciences  shows  that  the  slow  and  laborious 
path  of  induction  has  not  always  been  that  of  inventors. 
The  imagination,  impatient  to  arrive  at  the  causes, 
takes  pleasure  in  creating  hypotheses,  and  often  it 
changes  the  facts  in  order  to  adapt  them  to  its  work ; 
then  the  hypotheses  are  dangerous.  But  when  one 
regards  them  only  as  the  means  of  connecting  the 
phenomena  in  order  to  discover  the  laws;  when,  by 
refusing  to  attribute  them  to  a  reality,  one  rectifies 
them  continually  by  new  observations,  they  are  able 
to  lead  to  the  veritable  causes,  or  at  least  put  us  in  a 
position  to  conclude  from  the  phenomena  observed 
those  which  given  circumstances  ought  to  produce. 

If  we  should  try  all  the  hypotheses  which  can  be 
formed  in  regard  to  the  cause  of  phenomena  we  should 
arrive,  by  a  process  of  exclusion,  at  the  true  one. 
This  means  has  been  employed  with  success ;  some- 
times we  have  arrived  at  several  hypotheses  which 
explain  equally  well  all  the  facts  known,  and  among 
which  scholars  are  divided,  until  decisive  observations 
have  made  known  the  true  one.  Then  it  is  interesting, 
for  the  history  of  the  human  mind,  to  return  to  these 
hypotheses,  to  see  how  they  succeed  in  explaining  a 
great  number  of  facts,  and  to  investigate  the  changes 
which  they  ought  to  undergo  in  order  to  agree  with  the 
history  of  nature.  It  is  thus  that  the  system  01 
Ptolemy,  which  is  only  the  realization  of  celestial 


1 84     A  PHILOSOPHICAL  ESSAY  ON  PROBABILITIES. 

appearances,  is  transformed  into  the  hypothesis  of  the 
movement  of  the  planets  about  the  sun,  by  rendering 
equal  and  parallel  to  the  solar  orbit  the  circles  and  the 
epicycles  which  he  causes  to  be  described  annually, 
and  the  magnitude  of  which  he  leaves  undetermined. 
It  suffices,  then,  in  order  to  change  this  hypothesis  into 
the  true  system  of  the  world,  to  transport  the  apparent 
movement  of  the  sun  in  a  sense  contrary  to  the  earth. 

It  is  almost  always  impossible  to  submit  to  calculus 
the  probability  of  the  results  obtained  by  these  various 
means ;  this  is  true  likewise  for  historical  facts.  But 
the  totality  of  the  phenomena  explained,  or  of  the 
testimonies,  is  sometimes  such  that  without  being  able 
to  appreciate  the  probability  we  cannot  reasonably 
permit  ourselves  any  doubt  in  regard  to  them.  In  the 
other  cases  it  is  prudent  to  admit  them  only  with  great 
reserve. 


CHAPTER    XVIII. 

HISTORICAL  NOTICE  CONCERNING   THE  CAL- 
CULUS OF  PROBABILITIES. 

LONG  ago  were  determined,  in  the  simplest  games, 
the  ratios  of  the  chances  which  are  favorable  or 
unfavorable  to  the  players;  the  stakes  and  the  bets 
were  regulated  according  to  these  ratios.  But  no  one 
before  Pascal  and  Fermat  had  given  the  principles  and 
the  methods  for  submitting  this  subject  to  calculus,  and 
no  one  had  solved  the  rather  complicated  questions  of 
this  kind.  It  is,  then,  to  these  two  great  geometricians 
that  we  must  refer  the  first  elements  of  the  science  of 
probabilities,  the  discovery  of  which  can  be  ranked 
among  the  remarkable  things  which  have  rendered 
illustrious  the  seventeenth  century — the  century  which 
has  done  the  greatest  honor  to  the  human  mind.  The 
principal  problem  which  they  solved  by  different 
methods,  consists,  as  we  have  seen,  in  distributing 
equitably  the  stake  among  the  players,  who  are  sup- 
posed to  be  equally  skilful  and  who  agree  to  stop  the 
game  before  it  is  finished,  the  condition  of  play  being 
that,  in  order  to  win  the  game,  one  must  gain  a  given 
number  of  points  different  for  each  of  the  players.  It 

185 


1 86      A  PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

is  clear  that  the  distribution  should  be  made  propor- 
tionally to  the  respective  probabilities  of  the  players  of 
winning  this  game,  the  probabilities  depending  upon 
the  numbers  of  points  which  are  still  lacking.  The 
method  of  Pascal  is  very  ingenious,  and  is  at  bottom 
only  the  equation  of  partial  differences  of  this  problem 
applied  in  determining  the  successive  probabilities  of 
the  players,  by  going  from  the  smallest  numbers  to  the 
following  ones.  This  method  is  limited  to  the  case  of 
two  players;  that  of  Fermat,  based  upon  combinations, 
applies  to  any  number  of  players.  Pascal  believed  at 
first  that  it  was,  like  his  own,  restricted  to  two  players; 
this  brought  about  between  them  a  discussion,  at  the 
conclusion  of  which  Pascal  recognized  the  generality 
of  the  method  of  Fermat. 

Huygens  united  the  divers  problems  which  had 
already  been  solved  and  added  new  ones  in  a  little 
treatise,  the  first  that  has  appeared  on  this  subject 
and  which  has  the  title  De  Ratiociniis  in  ludo  alece. 
Several  geometricians  have  occupied  themselves  with 
the  subject  since:  Hudde,  the  great  pensionary,  Witt 
in  Holland,  and  Halley  in  England,  applied  calculus 
to  the  probabilities  of  human  life,  and  Halley  published 
in  this  field  the  first  table  of  mortality.  About  the 
same  time  Jacques  Bernoulli  proposed  to  geometricians 
various  problems  of  probability,  of  which  he  afterwards 
gave  solutions.  Finally  he  composed  his  beautiful 
work  entitled  Ars  conjcctandi,  which  appeared  seven 
years  after  his  death,  which  occurred  in  1706.  The 
science  of  probabilities  is  more  profoundly  investigated 
in  this  work  than  in  that  of  Huygens.  The  author 
gives  a  general  theory  of  combinations  and  series,  and 


THE    CALCULUS   OF  PROBABILITIES.  187 

applies  it  to  several  difficult  questions  concerning 
hazards.  This  work  is  still  remarkable  on  account  of 
the  justice  and  the  cleverness  of  view,  the  employment 
of  the  formula  of  the  binomial  in  this  kind  of  questions, 
and  by  the  demonstration  of  this  theorem,  namely, 
that  in  multiplying  indefinitely  the  observations  and 
the  experiences,  the  ratio  of  the  events  of  different 
natures  approaches  that  of  their  respective  probabilities 
in  the  limits  whose  interval  becomes  more  and  more 
narrow  in  proportion  as  they  are  multiplied,  and 
become  less  than  any  assignable  quantity.  This 
theorem  is  very  useful  for  obtaining  by  observations 
the  laws  and  the  causes  of  phenomena.  Bernoulli 
attaches,  with  reason,  a  great  importance  to  his  demon- 
stration, upon  which  he  has  said  to  have  meditated  for 
twenty  years. 

In  the  interval,  from  the  death  of  Jacques  Bernoulli 
to  the  publication  of  his  work,  Montmort  and  Moivre 
produced  two  treatises  upon  the  calculus  of  probabili- 
ties. That  of  Montmort  has  the  title  Ess  at  sur  les 
Jeux  de  hasard;  it  contains  numerous  applications  of 
this  calculus  to  various  games.  The  author  has  added 
in  the  second  edition  some  letters  in  which  Nicolas 
Bernoulli  gives  the  ingenious  solutions  of  several  diffi- 
cult problems.  The  treatise  of  Moivre,  later  than  that 
of  Montmort,  appeared  at  first  in  the  Transactions 
pliilosopliiqucs  of  the  year  1711.  Then  the  author 
published  it  separately,  and  he  has  improved  it  succes- 
sively in  three  editions.  This  work  is  principally  based 
upon  the  formula  of  the  binomial  and  the  problems 
which  it  contains  have,  like  their  solutions,  a  grand 
generality.  But  its  distinguishing  feature  is  the  theory 


1 88      A  PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

of  recurrent  series  and  their  use  in  this  subject.  This 
theory  is  the  integration  of  linear  equations  of  finite 
differences  with  constant  coefficients,  which  Moivre 
made  in  a  very  happy  manner. 

In  his  work,  Moivre  has  taken  up  again  the  theory 
of  Jacques  Bernoulli  in  regard  to  the  probability  of 
results  determined  by  a  great  number  of  observations. 
He  does  not  content  himself  with  showing,  as  Bernoulli 
does,  that  the  ratio  of  the  events  which  ought  to  occur 
approaches  without  ceasing  that  of  their  respective 
probabilities;  but  he  gives  besides  an  elegant  and 
simple  expression  of  the  probability  that  the  difference 
of  these  two  ratios  is  contained  within  the  given  limits. 
For  this  purpose  he  determines  the  ratio  of  the  greatest 
term  of  the  development  of  a  very  high  power  of  the 
binomial  to  the  sum  of  all  its  terms,  and  the  hyperbolic 
logarithm  of  the  excess  of  this  term  above  the  terms 
adjacent  to  it. 

The  greatest  term  being  then  the  product  of  a  con- 
siderable number  of  factors,  his  numerical  calculus 
becomes  impracticable.  In  order  to  obtain  it  by  a 
convergent  approximation,  Moivre  makes  use  of  a 
theorem  of  Stirling  in  regard  to  the  mean  term  of  the 
binomial  raised  to  a  high  power,  a  remarkable 
theorem,  especially  in  this,  that  it  introduces  the  square 
root  of  the  ratio  of  the  circumference  to  the  radius  in 
an  expression  which  seemingly  ought  to  be  irrelevant 
to  this  transcendent.  Moreover,  Moivre  was  greatly 
struck  by  this  result,  which  Stirling  had  deduced  from 
the  expression  of  the  circumference  in  infinite  products ; 
Wallis  had  arrived  at  this  expression  by  a  singlar 


THE   CALCULUS  OF  PROBABILITIES.  189 

analysis  which  contains  the  germ  of  the  very  curious 
and  useful  theory  of  definite  intergrals. 

Many  scholars,  among  whom  one  ought  to  name 
Deparcieux,  Kersseboom,  Wargentin,  Dupre  de  Saint- 
Maure,  Simpson,  Sussmilch,  Messene,  Moheau,  Price, 
Bailey,  and  Duvillard,  have  collected  a  great  amount 
of  precise  data  in  regard  to  population,  births,  mar- 
riages, and  mortality.  They  have  given  formulae  and 
tables  relative  to  life  annuities,  tontines,  assurances, 
etc.  But  in  this  short  notice  I  can  only  indicate  these 
useful  works  in  order  to  adhere  to  original  ideas.  Of 
this  number  special  mention  is  due  to  the  mathematical 
and  moral  hopes  and  to  the  ingenious  principle  which 
Daniel  Bernoulli  has  given  for  submitting  the  latter  to 
analysis.  Such  is  again  the  happy  application  which 
he  has  made  of  the  calculus  of  probabilities  to  inocula- 
tion. One  ought  especially  to  include,  in  the  number 
of  these  original  ideas,  direct  consideration  of  the 
possibility  of  events  drawn  from  events  observed. 
Jacques  Bernoulli  and  Moivre  supposed  these  possibili- 
ties known,  and  they  sought  the  probability  that  the 
result  of  future  experiences  will  more  and  more  nearly 
represent  them.  Bayes,  in  the  Transactions  pliiloso- 
phiqncs  of  the  year  1763,  sought  directly  the  probability 
that  the  possibilities  indicated  by  past  experiences  are 
comprised  within  given  limits;  and  he  has  arrived  at 
this  in  a  refined  and  very  ingenious  manner,  although 
a  little  perplexing.  This  subject  is  connected  with  the 
theory  of  the  probability  of  causes  and  future  events, 
concluded  from  events  observed.  Some  years  later  I 
expounded  the  principles  of  this  theory  with  a  remark 
as  to  the  influence  of  the  inequalities  which  may  exist 


T9°      A  PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

among  the  chances  which  are  supposed  to  be  equal. 
Although  it  is  not  known  which  of  the  simple  events 
these  inequalities  favor,  nevertheless  this  ignorance 
itself  often  increases  the  probability  of  compound 
events. 

In  generalizing  analysis  and  the  problems  concern- 
ing probabilities,  I  was  led  to  the  calculus  of  partial 
finite  differences,  which  Lagrange  has  since  treated  by 
a  very  simple  method,  elegant  applications  of  which 
he  has  used  in  this  kind  of  problems.  The  theory  of 
generative  functions  which  I  published  about  the  same 
time  includes  these  subjects  among  those  it  embraces, 
and  is  adapted  of  itself  and  with  the  greatest  generality 
to  the  most  difficult  questions  of  probability.  It  deter- 
mines again,  by  very  convergent  approximations,  the 
values  of  the  functions  composed  of  a  great  number  of 
terms  and  factors ;  and  in  showing  that  the  square  root 
of  the  ratio  of  the  circumference  to  the  radius  enters 
most  frequently  into  these  values,  it  shows  that  an 
infinity  of  other  transcendents  may  be  introduced. 

Testimonies,  votes,  and  the  decisions  of  electoral 
and  deliberative  assemblies,  and  the  judgments  of 
tribunals,  have  been  submitted  likewise  to  the  calculus 
of  probabilities.  So  many  passions,  divers  interests, 
and  circumstances  complicate  the  questions  relative  to 
the  subjects,  that  they  are  almost  always  insoluble. 
But  the  solution  of  very  simple  problems  which  have  a 
great  analogy  with  them,  may  often  shed  upon  difficult 
and  important  questions  great  light,  which  the  surety 
of  calculus  renders  always  preferable  to  the  most 
specious  reasonings. 

One  of  the  most  interesting  applications  of  the  cal- 


THE   CALCULUS   OF  PROBABILITIES.  191 

culus  of  probabilities  concerns  the  mean  values  which 
must  be  chosen  among  the  results  of  observations. 
Many  geometricians  have  studied  the  subject,  and 
Lagrange  has  published  in  the  Memoircs  de  Turin  a 
beautiful  method  for  determining  these  mean  values 
when  the  law  of  the  errors  of  the  observations  is 
known.  I  have  given  for  the  same  purpose  a  method 
based  upon  a  singular  contrivance  which  may  be 
employed  with  advantage  in  other  questions  of  analysis; 
and  this,  by  permitting  indefinite  extension  in  the 
whole  course  of  a  long  calculation  of  the  functions 
which  ought  to  be  limited  by  the  nature  of  the 
problem,  indicates  the  modifications  which  each  term 
of  the  final  result  ought  to  receive  by  virtue  of  these 
limitations.  It  has  already  been  seen  that  each 
observation  furnishes  an  equation  of  condition  of  the 
first  degree,  which  may  always  be  disposed  of  in  such 
a  manner  that  all  its  terms  be  in  the  first  member,  the 
second  being  zero.  The  use  of  these  equations  is  one 
of  the  principal  causes  of  the  great  precision  of  our 
astronomical  tables,  because  an  immense  number  of 
excellent  observations  has  thus  been  made  to  concur 
in  determining  their  elements.  When  there  is  only 
one  element  to  be  determined  Cotes  prescribed  that 
the  equations  of  condition  should  be  prepared  in  such 
a  manner  that  the  coefficient  of  the  unknown  element 
be  positive  in  each  of  them ;  and  that  all  these  equa- 
tions should  be  added  in  order  to  form  a  final  equation, 
whence  is  derived  the  value  of  this  element.  The  rule 
of  Cotes  was  followed  by  all  calculators,  but  since  he 
failed  to  determine  several  elements,  there  \vas  no  fixed 
rule  for  combining  the  equations  of  condition  in  such  a 


I92      A  PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

manner  as  to  obtain  the  necessary  final  equations;  but 
one  chose  for  each  element  the  observations  most  suit- 
able to  determine  it.  It  was  in  order  to  obviate  these 
gropings  that  Legendre  and  Gauss  concluded  to  add 
the  squares  of  the  first  members  of  the  equations  of 
condition,  and  to  render  the  sum  a  minimum,  by  vary- 
ing each  unknown  element;  by  this  means  is  obtained 
directly  as  many  final  equations  as  there  are  elements. 
But  do  the  values  determined  by  these  equations  merit 
the  preference  over  all  those  which  may  be  obtained 
by  other  means  ?  This  question,  the  calculus  of  prob- 
abilities alone  was  able  to  answer.  I  applied  it,  then, 
to  this  subject,  and  obtained  by  a  delicate  analysis  a 
rule  which  includes  the  preceding  method,  and  which 
adds  to  the  advantage  of  giving,  by  a  regular  process, 
the  desired  elements  that  of  obtaining  them  with  the 
greatest  show  of  evidence  from  the  totality  of  observa- 
tions, and  of  determining  the  values  which  leave  only 
the  smallest  possible  errors  to  be  feared. 

However,  we  have  only  an  imperfect  knowledge  of 
the  results  obtained,  as  long  as  the  law  of  the  errors 
of  which  they  are  susceptible  is  unknown;  we  must  be 
able  to  assign  the  probability  that  these  errors  are 
contained  within  given  limits,  which  amounts  to  deter- 
mining that  which  I  have  called  the  weight  of  a  result. 
Analysis  leads  to  general  and  simple  formulae  for  this 
purpose.  I  have  applied  this  analysis  to  the  results  of 
geodetic  observations.  The  general  problem  consists 
in  determining  the  probabilities  that  the  values  of  one 
or  of  several  linear  functions,  of  the  errors  of  a  very 
great  number  of  observations  are  contained  within  any 
limits, 


THE  CALCULUS   OF  PROBABILITIES.  193 

The  law  of  the  possibility  of  the  errors  of  observa- 
tions introduces  into  the  expressions  of  these  prob- 
abilities a  constant,  whose  value  seems  to  require  the 
knowledge  of  this  law,  which  is  almost  always 
unknown.  Happily  this  constant  can  be  determined 
from  the  observations. 

In  the  investigation  of  astronomical  elements  it  is 
given  by  the  sum  of  the  squares  of  the  differences 
between  each  observation  and  the  calculated  one. 
The  errors  equally  probable  being  proportional  to  the 
square  root  of  this  sum,  one  can,  by  the  comparison  of 
these  squares,  appreciate  the  relative  exactitude  of  the 
different  tables  of  the  same  star.  In  geodetic  opera- 
tions these  squares  are  replaced  by  the  squares  of  the 
errors  of  the  sums  observed  of  the  three  angles  of  each 
triangle.  The  comparison  of  the  squares  of  these 
errors  will  enable  us  to  judge  of  the  relative  precision 
of  the  instruments  with  which  the  angles  have  been 
measured.  By  this  comparison  is  seen  the  advantage 
of  the  repeating  circle  over  the  instruments  which  it 
has  replaced  in  geodesy. 

There  often  exists  in  the  observations  many  sources 
of  errors :  thus  the  positions  of  the  stars  being  deter- 
mined by  means  of  the  meridian  telescope  and  of  the 
circle,  both  susceptible  of  errors  whose  law  of  prob- 
ability ought  not  to  be  supposed  the  same,  the  elements 
that  are  deduced  from  these  positions  are  affected  by 
these  errors.  The  equations  of  condition,  which  are 
made  to  obtain  these  elements,  contain  the  errors  of 
each  instrument  and  they  have  various  coefficients. 
The  most  advantageous  system  of  factors  by  which 
these  equations  ought  to  be  multiplied  respectively,  in 


194      A  PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

order  to  obtain,  by  the  union  of  the  products,  as  many 
final  equations  as  there  are  elements  to  be  determined, 
is  no  longer  that  of  the  coefficients  of  the  elements  in 
each  equation  of  condition.  The  analysis  which  I  have 
used  leads  easily,  whatever  the  number  of  the  sources 
of  error  may  be,  to  the  system  of  factors  which  gives 
the  most  advantageous  results,  or  those  in  which  the 
same  error  is  less  probable  than  in  any  other  system. 
The  same  analysis  determines  the  laws  of  probability 
of  the  errors  of  these  results.  These  formulae  contain 
as  many  unknown  constants  as  there  are  sources  of 
error,  and  they  depend  upon  the  laws  of  probability  of 
these  errors.  It  has  been  seen  that,  in  the  case  of  a 
single  source,  this  constant  can  be  determined  by 
forming  the  sum  of  the  squares  of  the  residuals  of  each 
equation  of  condition,  when  the  values  found  for  these 
elements  have  been  substituted.  A  similar  process 
generally  gives  values  of  these  constants,  whatever 
their  number  may  be,  which  completes  the  application 
of  the  calculus  of  probabilities  to  the  results  of  observa- 
tions. 

I  ought  to  make  here  an  important  remark.  The 
small  uncertainty  that  the  observations,  when  they  are 
not  numerous,  leave  in  regard  to  the  values  of  the 
constants  of  which  I  have  just  spoken,  renders  a  little 
uncertain  the  probabilities  determined  by  analysis. 
But  it  almost  always  suffices  to  know  if  the  probability, 
that  the  errors  of  the  results  obtained  are  comprised 
within  narrow  limits,  approaches  closely  to  unity;  and 
when  it  is  not,  it  suffices  to  know  up  to  what  point  the 
observations  should  be  multiplied,  in  order  to  obtain  a 
probability  such  that  no  reasonable  doubt  remains  in 


THE   CALCULUS  OF  PROBABILITIES.  195 

regard  to  the  correctness  of  the  results.  The  analytic 
formulae  of  probabilities  satisfy  perfectly  this  require- 
ment; and  in  this  connection  they  may  be  viewed  as 
the  necessary  complement  of  the  sciences,  based  upon 
a  totality  of  observations  susceptible  of  error.  They 
are  likewise  indispensable  in  solving  a  great  number  of 
problems  in  the  natural  and  moral  sciences.  The 
regular  causes  of  phenomena  are  most  frequently  either 
unknown,  or  too  complicated  to  be  submitted  to  cal- 
culus; again,  their  action  is  often  disturbed  by  accidental 
and  irregular  causes;  but  its  impression  always  remains 
in  the  events  produced  by  all  these  causes,  and  it  leads 
to  modifications  which  only  a  long  series  of  observa- 
tions can  determine.  The  analysis  of  probabilities 
develops  these  modifications ;  it  assigns  the  probability 
of  their  causes  and  it  indicates  the  means  of  continually 
increasing  this  probability.  Thus  in  the  midst  of  the 
irregular  causes  which  disturb  the  atmosphere,  the 
periodic  changes  of  solar  heat,  from  day  to  night,  and 
from  winter  to  summer,  produce  in  the  pressure  of  this 
great  fluid  mass  and  in  the  corresponding  height  of  the 
barometer,  the  diurnal  and  annual  oscillations;  and 
numerous  barometric  observations  have  revealed  the 
former  with  a  probability  at  least  equal  to  that  of  the 
facts  which  we  regard  as  certain.  Thus  it  is  again 
that  the  series  of  historical  events  shows  us  the  con- 
stant action  of  the  great  principles  of  ethics  in  the 
midst  of  the  passions  and  the  various  interests  which 
disturb  societies  in  every  way.  It  is  remarkable  that 
a  science,  which  commenced  with  the  consideration  of 
games  of  chance,  should  be  elevated  to  the  rank  of  the 
most  important  subjects  of  human  knowlegdge. 


I96      A  PHILOSOPHICAL   ESSAY  ON  PROBABILITIES. 

I  have  collected  all  these  methods  in  my  TJieorie 
analytique  des  Probabilite's,  in  which  I  have  proposed 
to  expound  in  the  most  general  manner  the  principles 
and  the  analysis  of  the  calculus  of  probabilities,  like- 
wise the  solutions  of  the  most  interesting  and  most 
difficult  problems  which  calculus  presents. 

It  is  seen  in  this  essay  that  the  theory  of  probabilities 
is  at  bottom  only  common  sense  reduced  to  calculus; 
it  makes  us  appreciate  with  exactitude  that  which  exact 
minds  feel  by  a  sort  of  instinct  without  being  able 
ofttimes  to  give  a  reason  for  it.  It  leaves  no  arbitrari- 
ness in  the  choice  of  opinions  and  sides  to  be  taken ; 
and  by  its  use  can  always  be  determined  the  most 
advantageous  choice.  Thereby  it  supplements  most 
happily  the  ignorance  and  the  weakness  of  the  human 
mind.  If  we  consider  the  analytical  methods  to  which 
this  theory  has  given  birth ;  the  truth  of  the  principles 
which  serve  as  a  basis ;  the  fine  and  delicate  logic 
which  their  employment  in  the  solution  of  problems 
requires ;  the  establishments  of  public  utility  which  rest 
upon  it ;  the  extension  which  it  has  received  and  which 
it  can  still  receive  by  its  application  to  the  most  impor- 
tant questions  of  natural  philosophy  and  the  moral 
science ;  if  we  consider  again  that,  even  in  the  things 
which  cannot  be  submitted  to  calculus,  it  gives  the 
surest  hints  which  can  guide  us  in  our  judgments,  and 
that  it  teaches  us  to  avoid  the  illusions  which  ofttimes 
confuse  us,  then  we  shall  see  that  there  is  no  science 
more  worthy  of  our  meditations,  and  that  no  more 
useful  one  could  be  incorporated  in  the  system  of  public 
instruction. 


SHORT-TITLE     CATALOGUE 

OF  THE 

PUBLICATIONS 

OF 

JOHN   WILEY   &    SONS, 

NEW    YORK. 
LOJSDOX:    CITAPMAX  &  HALL.  LIMITED. 


ARRANGED  UNDER  SUBJECTS. 


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